An algebraic version of the Strong Black Box

Various versions of the prediction principle called the “Black Box” are known. One of the strongest versions can be found in [EM]. There it is formulated and proven in a model theoretic way. In order to apply it to specific algebraic problems it thus has to be transformed into the desired algebr...

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Published in:	Algebra and Discrete Mathematics
Date:	2003
Main Authors:	Göbel, R., Wallutis, S.L.
Format:	Article
Language:	English
Published:	Інститут прикладної математики і механіки НАН України 2003
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Cite this:	An algebraic version of the Strong Black Box / R. Göbel, S.L. Wallutis // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 7–45. — Бібліогр.: 9 назв. — англ.

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spelling	Göbel, R. Wallutis, S.L. 2019-06-17T10:48:19Z 2019-06-17T10:48:19Z 2003 An algebraic version of the Strong Black Box / R. Göbel, S.L. Wallutis // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 7–45. — Бібліогр.: 9 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 03E75, 20K20, 20K30; 13C99. https://nasplib.isofts.kiev.ua/handle/123456789/155695 Various versions of the prediction principle called the “Black Box” are known. One of the strongest versions can be found in [EM]. There it is formulated and proven in a model theoretic way. In order to apply it to specific algebraic problems it thus has to be transformed into the desired algebraic setting. This requires intimate knowledge on model theory which often prevents algebraists to use this powerful tool. Hence we here want to present algebraic versions of this “Strong Black Box” in order to demonstrate that the proofs are straightforward and that it is easy enough to change the setting without causing major changes in the relevant proofs. This shall be done by considering three different applications where the obtained results are actually known. The first author was supported by GIF project I-706-54.6/2001 of the GermanIsraeli Foundation for Scientific Research and Development. The second author was supported by the Deutsche Forschungsgemeinschaft and the Lise-Meitner-Programm des Ministeriums für Wissenschaft und Forschung NRW. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics An algebraic version of the Strong Black Box Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	An algebraic version of the Strong Black Box
spellingShingle	An algebraic version of the Strong Black Box Göbel, R. Wallutis, S.L.
title_short	An algebraic version of the Strong Black Box
title_full	An algebraic version of the Strong Black Box
title_fullStr	An algebraic version of the Strong Black Box
title_full_unstemmed	An algebraic version of the Strong Black Box
title_sort	algebraic version of the strong black box
author	Göbel, R. Wallutis, S.L.
author_facet	Göbel, R. Wallutis, S.L.
publishDate	2003
language	English
container_title	Algebra and Discrete Mathematics
publisher	Інститут прикладної математики і механіки НАН України
format	Article
description	Various versions of the prediction principle called the “Black Box” are known. One of the strongest versions can be found in [EM]. There it is formulated and proven in a model theoretic way. In order to apply it to specific algebraic problems it thus has to be transformed into the desired algebraic setting. This requires intimate knowledge on model theory which often prevents algebraists to use this powerful tool. Hence we here want to present algebraic versions of this “Strong Black Box” in order to demonstrate that the proofs are straightforward and that it is easy enough to change the setting without causing major changes in the relevant proofs. This shall be done by considering three different applications where the obtained results are actually known.
issn	1726-3255
url	https://nasplib.isofts.kiev.ua/handle/123456789/155695
citation_txt	An algebraic version of the Strong Black Box / R. Göbel, S.L. Wallutis // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 7–45. — Бібліогр.: 9 назв. — англ.
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first_indexed	2025-11-27T02:29:23Z
last_indexed	2025-11-27T02:29:23Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2003). pp. 7 – 45 c© Journal “Algebra and Discrete Mathematics” An algebraic version of the Strong Black Box Rüdiger Göbel and Simone L. Wallutis Communicated by D. Simson Abstract. Various versions of the prediction principle called the “Black Box” are known. One of the strongest versions can be found in [EM]. There it is formulated and proven in a model theoretic way. In order to apply it to specific algebraic problems it thus has to be transformed into the desired algebraic setting. This requires intimate knowledge on model theory which often pre- vents algebraists to use this powerful tool. Hence we here want to present algebraic versions of this “Strong Black Box” in order to demonstrate that the proofs are straightforward and that it is easy enough to change the setting without causing major changes in the relevant proofs. This shall be done by considering three different applications where the obtained results are actually known. Introduction The aim of this paper is to investigate and apply a well-known prediction principle due to Saharon Shelah. It can be found in [EM, Chapter XIII] where it is formulated and proven in a model theoretic way. However, many known applications are in an algebraic context and thus it seems natural to transfer everything into the corresponding set- ting. An algebraic version of this principle, which we call “Strong Black Box”, has the advantage of immediate application. Moreover, the here The first author was supported by GIF project I-706-54.6/2001 of the German- Israeli Foundation for Scientific Research and Development. The second author was supported by the Deutsche Forschungsgemeinschaft and the Lise-Meitner-Programm des Ministeriums für Wissenschaft und Forschung NRW. 2000 Mathematics Subject Classification: 03E75, 20K20, 20K30; 13C99. Key words and phrases: prediction principle, Black Box, endomorphism alge- bra, E-ring, E(R)-algebra, ultra-cotorsion-free module. Jo u rn al A lg eb ra D is cr et e M at h .8 Strong Black Box presented proofs are given in an algebraic setting and use only basic knowledge of set theory. Therefore it is relatively easy to adjust a given Strong Black Box to a different situation. To emphasize this we here present three versions and its applications. This shall be done in three independent sections: In the first section we demonstrate how to real- ize endomorphism rings. In Section 2 we construct E-rings, respectively E(R)-algebras, and in Section 3 we show the existence of a cotorsion-free module G with the additional property that, for any submodule H of G, if G/H is cotorsion-free then H = G or \|H\| < \|G\|; such a module G is called ultra-cotorsion-free. Note, the reader who is only interested in one of the applications can go straight to the favorite section; it will be clear if something from before is needed. We need to mention that the obtained results in all these applications are actually known. The classical proofs, however, are much more com- plicated due to the fact that they are based on the “General Black Box”. We want to use the name “General Black Box” for the Black Box principle as, for example, given in [S] or [CG] to distinguish between this principle and the one presented here. Both principles hold in ordinary set theory, ZFC, but are inspired by the diamond principle due to R. Jensen which holds in the constructible universe (V=L). While the Strong Black Box provides predictions on particular successor cardinals the General Black Box includes more general cardinals, even some singular ones. On the other hand, the Strong Black Box is less complicated and its applications more straightforward. This advantage is due to the fact that the predic- tion principle is sharper, i.e. the disjointness condition is stronger, and hence less algebra is needed. The reader who is familiar with constructions using the General Black Box will appreciate the achieved simplifications. For unexplained terminology we refer the reader to [EM]. §1. Realizing endomorphism rings Throughout this first section let R be a commutative ring with 1 and let S be a countable multiplicatively closed subset of R containing no units except 1 such that R is S-reduced and S-torsion-free. Recall, an R-module M is said to be S-reduced if it satisfies ⋂ s∈S sM = 0; M is S-torsion-free if sm = 0 (s ∈ S, m ∈ M) implies m = 0. Note that, in general, we shall skip the prefix “S-” and use the notions torsion-free, reduced, pure etc. instead of S-torsion-free, S-reduced, S-pure where M is said to be (S-) pure in N (notation: M ⊆∗ N) if sM = M ∩ sN holds for all s ∈ S. Hence, in this case, the S-adic topology on M is induced by the S-adic topology on N . Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 9 To describe the S-adic topology with a descending chain of submod- ules we fix an enumeration S = {sn\|n < ω} of S with s0 = 1 and define a divisor chain (qn)n<ω by qn = s0 · . . . · sn. Then the S-adic topology of any R-module M has {qnM \|n < ω} as a basis of neighbourhoods of zero. Let R̂ denote the completion of R in the S-adic topology. We shall assume that R satisfies HomR(R̂, R) = 0, that is to say R is S-cotorsion- free. In general, an R-module M is said to be (S-)cotorsion-free if M is (S-)torsion-free, (S-)reduced and satisfies HomR(R̂, M) = 0. Note, that the torsion-freeness of M actually follows from HomR(R̂, M) = 0 and in some cases, e.g. for domains R such that the quotient field Q(R) is countably generated over R, HomR(R̂, M) = 0 also implies that M is reduced. In this section we show that, given infinite cardinals κ, µ, λ satisfying κ ≥ \|R\| , µκ = µ, λ = µ+ and a cotorsion-free R-algebra A with F ⊆∗ A ⊆∗ F̂ for some free R-module F and \|A\| ≤ λ, there exists a cotorsion- free R-module G of cardinality λ such that EndRG = A. This shall be done using a suitable version of the Strong Black Box, which is introduced and proven in the first subsection. For a proof of this result using the General Black Box we refer to [CG]. 1.1. The Black Box Theorem In this subsection we shall formulate and prove the Strong Black Box in full detail. To do so let R, S, F, A as well as κ, µ, λ be as above. We formulate the parameters of the Black Box with respect to a free R-module B and its completion B̂. Let F = ⊕ ε<ρ Raε (⊆∗ A, ρ ≤ λ) and put B := ⊕ α<λ eαF ⊆∗ ⊕ α<λ eαA =: B′. Then, writing eα,ε for eαaε and using that R is commutative, we have B = ⊕ (α,ε)∈λ×ρ Reα,ε and B̂ = B̂′. For later use we put the lexicographic ordering on λ × ρ; since λ, ρ are ordinals λ × ρ is well ordered. For any g = (gα,εeα,ε)(α,ε)∈λ×ρ ∈ B̂ ⊆ ∏ (α,ε)∈λ×ρ R̂eα,ε we define the support of g by [g] = {(α, ε) ∈ λ × ρ\|gα,ε 6= 0} and the support of M ⊆ B̂ by [M ] = ⋃ g∈M [g]; note \|[g]\| ≤ ℵ0 for all g ∈ B̂. Moreover, we define the λ-support of g by [g]λ = {α ∈ λ\|∃ε ∈ ρ : (α, ε) ∈ [g]} ⊆ λ and the A-support of g by [g]A = {ε ∈ ρ\|∃α ∈ λ : (α, ε) ∈ [g]} ⊆ ρ ⊆ λ. Jo u rn al A lg eb ra D is cr et e M at h .10 Strong Black Box Note, eα,ε = eαaε where aε ∈ A, which explains the use of the notion “A-support”. Next we define a norm (λ-norm) on λ, respectively on B̂, by ‖{α}‖ = α + 1 for an ordinal α in λ, ‖M‖ = sup α∈M ‖{α}‖ for a subset M of λ and ‖g‖ = ‖ [g]λ ‖ for g ∈ B̂, i.e. ‖g‖ = min{β ∈ λ\| [g]λ ⊆ β}. Note, [g]λ ⊆ β holds iff g ∈ B̂′ β for B′ β = ⊕ α<β eαA. We also define an A-norm of g by ‖g‖A = ‖ [g]A ‖. For a subset M of B̂ the above definitions extend naturally, e.g. [M ]λ = ⋃ g∈M [g]λ = {α ∈ λ\|∃ε ∈ ρ : (α, ε) ∈ [M ]}. The reader who is only interested in the case A = R, i.e. in realizing a ring R as endomorphism ring of an R-module, can ignore the “ρ-”, respectively the “A-component”, and just work with B = ⊕ α<λ Reα (cf. §2). To formulate the Black Box we finally need to define canonical homo- morphisms which shall play a crucial role in the proof of the Black Box. For this we, once and for all, fix bijections hα : µ → α for all α with µ ≤ α < λ where we put hµ = idµ (this is possible since λ = µ+ and so \|α\| = \|µ\| = µ for all such αs). For technical reasons we also put hα = hµ for α < µ. Note, Imhα = µ ∪ α for all α < λ. Definition 1.1.1. Let the bijections hα (α < λ) be as above and put hα,ε = hα × hε : µ × µ → Imhα × Imhε for all (α, ε) ∈ λ × ρ. We define P to be a canonical summand of B if P = ⊕ (α,ε)∈I Reα,ε for some I ⊆ λ × ρ with \|I\| ≤ κ such that: • if (α, ε) ∈ I then (I ∩ (µ × µ)) hα,ε = I ∩ Imhα,ε; • if (α, ε) ∈ I then (ε, ε) ∈ I; and • if (α, ε) ∈ I, α ∈ ρ then (ε, α) ∈ I. Accordingly, ϕ : P → B̂ is said to be a canonical homomorphism if P is a canonical summand of B and Imϕ ⊆ P̂ ; we put [ϕ] = [P ], [ϕ]λ = [P ]λ and ‖ϕ‖ = ‖P‖. Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 11 Note that, by the above definition, a canonical summand P satisfies ‖P‖A ≤ ‖P‖. Also note, that we are mainly interested in canonical homomorphisms whose norm is a limit ordinal of cofinality ω; hence we introduce the notation λo := {α < λ\|cf(α) = ω}. Let C denote the set of all canonical homomorphisms. By assumption, µκ = µ and thus 2κ ≤ µ and λκ = λ (see [J, Ch. I, (6.18)]), which implies \|C\| ≤ λ since \|{I ⊆ λ × ρ\| \|I\| ≤ κ}\| = λκ and, for a fixed canonical summand P , ∣∣∣HomR(P, P̂ ) ∣∣∣ ≤ 2κ. Note, \|C\| = λ then follows from the Strong Black Box Theorem. We are now ready to formulate the main theorem of this subsection, i.e. the desired version of the Strong Black Box: Strong Black Box Theorem 1.1.2. Let κ, µ, λ be as before and let E ⊆ λo be a stationary subset of λ. Then there exists a family C∗ of canonical homomorphisms with the following properties: (1) If ϕ ∈ C∗ then ‖ϕ‖ ∈ E. (2) If ϕ, ϕ′ are two different elements of C∗ of the same norm α then ‖ [ϕ]λ ∩ [ϕ′]λ ‖ < α. (3) Prediction: For any R-homomorphism ψ : B → B̂ and for any subset I of λ × ρ with \|I\| ≤ κ the set {α ∈ E\|∃ϕ ∈ C∗ : ‖ϕ‖ = α, ϕ ⊆ ψ, I ⊆ [ϕ]} is stationary. To prove the above theorem we need further definitions and other results. We begin with defining an equivalence relation on C as follows: Definition 1.1.3. Canonical homomorphisms ϕ, ϕ′ are said to be equiv- alent, or of the same type (notation: ϕ ≡ ϕ′), if [ϕ] ∩ (µ × µ) = [ ϕ′ ] ∩ (µ × µ) and there exists an order-isomorphism f : [ϕ] → [ϕ′] such that (xf̄)ϕ′ = (xϕ)f̄ for all x ∈ domϕ where f̄ : d̂omϕ → d̂omϕ′ is the unique extension of the R-homomorphism defined by eα,εf̄ = e(α,ε)f ((α, ε) ∈ [ϕ]) . Jo u rn al A lg eb ra D is cr et e M at h .12 Strong Black Box Note, f : [ϕ] → [ϕ′] is unique since [ϕ] , [ϕ′] are well ordered. Thus, if ϕ ≡ ϕ′ and [ϕ] = [ϕ′] then f = id and so ϕ = ϕ′. Obviously, any type in (C,≡) can be represented by a subset V of µ×µ of cardinality at most κ, an order-type of a set of cardinality κ and a homomorphism from a free R-module P of rank κ into its completion P̂ . Therefore there are at most µ different types (equivalence classes) since \|{V ⊆ µ × µ\| \|V \| ≤ κ}\| = µκ = µ, there are at most 2κ ≤ µ non- isomorphic well-orderings on a set of size κ and, for a fixed P , we have∣∣∣HomR(P, P̂ ) ∣∣∣ = 2κ. Certain infinite sequences of canonical homomorphisms play an im- portant role: Definition 1.1.4. Let ϕ0 ⊂ ϕ1 ⊂ . . . ⊂ ϕn ⊂ . . . (n < ω) be an increas- ing sequence of canonical homomorphisms. Then (ϕn)n<ω is said to be admissible if [ϕ0] ∩ (µ × µ) = [ϕn] ∩ (µ × µ) and ‖ϕn‖ < ‖ϕn+1‖ for all n < ω. Also, we say that (ϕn)n<ω is admissible for a sequence (βn)n<ω of ordinals in λ (or (ϕn, βn)n<ω is admissible), if (ϕn)n<ω is admissible satisfying ‖ϕn‖ ≤ βn < ‖ϕn+1‖ and [ϕn] = [ϕn+1] ∩ (βn × βn) for all n < ω. Moreover, two admissible sequences (ϕn)n<ω, (ϕ′ n)n<ω are said to be equivalent, or of the same type, if ϕn ≡ ϕ′ n for all n < ω. Note, if (ϕn)n<ω is admissible then ϕ = ⋃ n<ω ϕn is an element of C with ‖ϕ‖ = supn<ω ‖ϕn‖ ∈ λo. Let T denote the set of all possible types of admissible sequences of canonical homomorphisms. It follows immediately from the above definition that any type in T can be identified with a sequence (τn)n<ω for some types τn in (C,≡) where the corresponding subsets of µ× µ are all the same. Hence we clearly deduce \|T\| ≤ µℵ0 = µ. If (ϕn)n<ω, respectively (ϕn, βn)n<ω, is admissible of type τ , then we also use the notion τ -admissible. Moreover, if τ = (τn)n<ω ∈ T and (ϕn)n<k (k < ω) is a finite increasing sequence of canonical homomor- phisms satisfying ϕn ∈ τn and ‖ϕn‖ < ‖ϕn+1‖ for all n < k, then we shall also speak of (ϕn)n<k to be of type τ , keeping in mind that such a finite sequence could belong to different types in T. We are now ready to show the following result which will play a crucial role in proving the Black Box Theorem 1.1.2. Note, that the kind Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 13 of formula we use to formulate this result goes under the generic name “Svenonius sentences” or “Svenonius game” (cf. [H, p.112]). It should be noted for the reader who is familiar with game theory that the proof below uses the Gale-Stewart-Theorem, namely that in a closed (or open) game some player has a winning strategy. Proposition 1.1.5. Let ψ : B → B̂ be an R-homomorphism, I ⊆ λ × ρ a set of cardinality at most κ and K = Kψ,I = {ϕ ∈ C\|ϕ ⊆ ψ, I ⊆ [ϕ]}. Then there exists a type τ ∈ T such that ∃ϕ0 ∈ K ∀β0 ≥ ‖ϕ0‖ . . .∃ϕn ∈ K ∀βn ≥ ‖ϕn‖ . . . with (ϕn, βn)n<ω being τ -admissible. Proof. Suppose, for contradiction, that the conclusion fails. Then, since the above formula is of “finite character”we have, for any type τ ∈ T, ∀ϕ0 ∈ K ∃β0(τ, ϕ0) ≥ ‖ϕ0‖ . . .∀ϕn ∈ K ∃βn(τ, ϕ0, . . . , ϕn) ≥ ‖ϕn‖ . . . with (ϕn, βn)n<ω not being τ -admissible. In the following we fix ordinals βn(τ, ϕ0, . . . , ϕn) as above (τ ∈ T, ϕi ∈ K, i ≤ n < ω). Moreover let Tα = α × (α ∩ ρ) and Bα = ⊕ (β,ε)∈Tα Reβ,ε (α < λ). We define C to be the set of all ordinals α < λ such that Bαψ ⊆ B̂α and βn(τ, ϕ0, . . . , ϕn) ≤ α for each type τ ∈ T and for any finite sequence (ϕ0, . . . , ϕn) with ϕi ∈ K and ‖ϕi‖ ≤ α (that is, iff [ϕi] ⊆ α × α) for all i ≤ n. Then C is unbounded since: Given an arbitrary α0 < λ we inductively define ordinals αk < λ (k < κ+ ≤ µ) by αk = sup{αl\|l < k} for k a limit ordinal and αk+1 = αk + 1 ∨ ‖Bαk ψ‖ ∨ sup Bk where Bk = {βn(τ, ϕ0, . . . , ϕn)\|τ ∈ T, ϕi ∈ K, ‖ϕi‖ ≤ αk} is a set of cardinality at most µ since \|T \| ≤ µ, \|{ϕ ∈ K\|‖ϕ‖ ≤ αk}\| ≤ \|αk\| κ ≤ µκ = µ provided that αk < λ. Then, using that \|[ϕ]\| ≤ κ for all ϕ ∈ K, it is easy to see that α = sup{αk\|k < κ+} is an element of C. Now we choose an increasing sequence α0 < α1 < . . . < αn < . . . (n < ω) in C with α0 ≥ µ, ‖I‖, ‖I‖A and put α = supn<ω αn. Note, since αn ∈ C for all n < ω we also have that Bαψ ⊆ B̂α. Moreover, let {εn\|n < ω} be an arbitrary but fixed set of elements of α ∩ ρ. Using these αns and Jo u rn al A lg eb ra D is cr et e M at h .14 Strong Black Box εns we inductively define subsets In of α × (α ∩ ρ) = Tα = [Bα] (n < ω) of cardinality at most κ by: I0 = I ∪ {(αn, εn)\|n < ω} and In+1 = In ∪ In ∪ In ψ ∪ In h where In = {(ε, ε)\|∃β ∈ λ : (β, ε) ∈ In} ∪ {(ε, β)\|∃β ∈ ρ : (β, ε) ∈ In}, In ψ = ⋃ (β,ε)∈In [eβ,εψ] , and In h = ⋃ (β,ε)∈In ( (In ∩ (µ × µ)) hβ,ε ∪ (In ∩ Imhβ,ε) h−1 β,ε ) . These Ins really satisfy the required conditions since: For n = 0 we have \|I0\| ≤ κ and I0 ⊆ Tα by ‖I‖, ‖I‖A ≤ α0 < α. Next suppose \|In\| ≤ κ and In ⊆ Tα. Then, clearly, \|In+1\| ≤ κ; also In ⊆ Tα is obvious, In ψ ⊆ Tα holds since Bαψ ⊆ B̂α, In h ⊆ Tα follows from the definition of the hβ,εs, and so In+1 ⊆ Tα as required. Now we put I∗ = ⋃ n<ω In and P = ⊕ (β,ε)∈I∗ Reβ,ε. Then \|I∗\| ≤ κ, I∗ ⊆ Tα = α × (α ∩ ρ) and I ⊆ I∗ = [P ]. Moreover, (I∗ ∩ (µ × µ)) hβ,ε = I∗ ∩ Imhβ,ε for all (β, ε) ∈ I∗ by the definition of the In h s and I∗ also satisfies: if (β, ε) ∈ I∗ then (ε, ε) ∈ I∗ and if (β, ε) ∈ I∗, β ∈ ρ then (ε, β) ∈ I∗ (see definition of the Ins). Therefore P is a canonical summand of B (see Definition 1.1.1) with Pψ ⊆ P̂ where the latter follows from the definition of the In ψ s. Thus ϕ := ψ ¹ P is a canonical homomorphism with I ⊆ I∗ = [ϕ], i.e. ϕ ∈ K. Finally, we put ϕn = ϕ ¹ (P ∩ Bαn), that is [ϕn] = [ϕ] ∩ (αn × αn). Using the definitions of the Bαns and of the set C ⊆ Cψ it is easy to check that ϕn ∈ K (n < ω) and that then (ϕn)n<ω is an admissible sequence with ‖ϕn‖ ≤ αn < ‖ϕn+1‖. Let τ ∈ T be the type of (ϕn)n<ω. By the definition of C we also have that βn(τ, ϕ0, . . . , ϕn) ≤ αn since ‖ϕn‖ ≤ αn for any n < ω. Therefore, ‖ϕn‖ ≤ βn ≤ αn and hence [ϕn] = [ϕ] ∩ (αn × αn) = [ϕ] ∩ (βn × βn) = [ϕn+1] ∩ (βn × βn), i.e. (ϕn, βn)n<ω is τ -admissible, contradicting the assumption that it is not for βn = βn(τ, ϕ0, . . . , ϕn). Hence the original conclusion holds and so the proof is finished. Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 15 In order to prove the Strong Black Box Theorem we also need the following known lemma. We include the proof for the convenience of the reader; it can also be found in [EM]. First recall that, for an ordinal α, a mapping ηα : ω → α is a ladder on α if it is strictly increasing and sup Imηα = α; an indexed family of such ladders on different αs is called a ladder system. Lemma 1.1.6. Let E ⊆ λo be a stationary subset of λ = µ+ for some µ such that µℵ0 = µ. Then there is a ladder system {ηα\|α ∈ E} such that, for all cubs C, the set {α ∈ E\|Imηα ⊆ C} is stationary. Proof. For any α ∈ E let {ηi α\|i < µ} be an enumeration of all ladders on α (if necessary with repetition); this is possible since \|ωα\| = \|α\|ℵ0 ≤ µℵ0 = µ for all α < λ = µ+. Moreover, for each i < µ, let ηi be the ladder system given by ηi := {ηi α\|α ∈ E}. We claim that there is an i < µ such that ηi satisfies the conclusion of the theorem. Suppose not. Then, for any i < µ, there is a cub Ci ⊆ λ such that the set Ti := {α ∈ E\|Imηi α ⊆ Ci} is not stationary, i.e. there is a cub Di with Ti ∩ Di = ∅. Replacing Ci by the cub Ci ∩ Di we may assume that Ti = ∅ for any i < µ, i.e. Imηi α 6⊆ Ci for all α ∈ E (i < µ). We put C = ⋂ i<µ Ci. Then C is also a cub in λ (cf. [EM, II, Proposition 4.3]). We choose an ordinal α ∈ C ∩ E which is a limit point of C, i.e. α = supn<ω αn for some αn ∈ C ∩ α with αn < αn+1 (α, respectively the αns, exist since the set of all limit points of a cub is also a cub; see [EM, p.35]). Therefore the map ηα : ω → α defined by ηα(n) = αn is a ladder on α with Imηα ⊆ C. By the above enumeration ηα = ηi α for some i < µ contradicting Imηi α 6⊆ Ci ⊇ C. Finally, we prove the main theorem of this subsection. Proof of the Strong Black Box Theorem 1.1.2. First we decom- pose the given stationary set E into \|T\| ≤ µ pairwise disjoint station- ary subsets, say E = ⋃ τ∈T Eτ . For each τ ∈ T we choose a ladder system {ηα\|α ∈ Eτ} such that the set {α ∈ Eτ \|Imηα ⊆ C} is stationary for any cub C (cf. Lemma 1.1.6). For any α ∈ Eτ , we define Cα ⊆ C to be the set of all canonical homomorphisms ϕ such that ‖ϕ‖ = α and ϕ = ⋃ n<ω ϕn for some τ - admissible sequence (ϕn)n<ω with [ϕn] = [ϕ] ∩ (ηα(n) × ηα(n)) (n < ω). Note, for ϕ, ϕ′ ∈ Cα with domϕ = domϕ′ (iff [ϕ] = [ϕ′]), we clearly deduce ϕn = ϕ′ n for all n < ω and so ϕ = ϕ′ (cf. Definition 1.1.3). Now we define C∗ to be the union of all these Cαs, i.e. C∗ = ⋃ α∈E Cα. First note that condition (1) obviously holds. Jo u rn al A lg eb ra D is cr et e M at h .16 Strong Black Box Next we show that condition (2) is satisfied. To do so let ϕ, ϕ′ ∈ C∗ with ‖ϕ‖ = ‖ϕ′‖ = α. Then ϕ, ϕ′ ∈ Cα where α ∈ Eτ for some τ ∈ T and so ϕ = ⋃ n<ω ϕn, ϕ′ = ⋃ n<ω ϕ′ n for some τ -admissible sequences (ϕn)n<ω, (ϕ′ n)n<ω. Suppose that ‖ [ϕ]λ ∩ [ϕ′]λ ‖ = α. Then there are αn ∈ [ϕ]λ ∩ [ϕ′]λ with supn<ω αn = α; w.l.o.g. we may assume αn ≥ µ. Let εn, ε′n ∈ ρ such that (αn, εn) ∈ [ϕ] and (αn, ε′n) ∈ [ϕ′]. We consider two cases. Firstly assume that ρ < λ, i.e. ρ ≤ µ. Then hαn,εn = hαn × hµ = hαn,ε′n (n < ω). Since (ϕn)n<ω, (ϕ′ n)n<ω are of the same type τ we know that [ϕ] ∩ (µ × µ) = [ϕ0] ∩ (µ × µ) = [ ϕ′ 0 ] ∩ (µ × µ) = [ ϕ′ ] ∩ (µ × µ). Hence [ϕ] ∩ (αn × µ) = [ϕ] ∩ Imhαn,εn = = ([ϕ] ∩ (µ × µ)) hαn,εn = ([ ϕ′ ] ∩ (µ × µ) ) hαn,ε′n = = [ ϕ′ ] ∩ Imhαn,ε′n = [ ϕ′ ] ∩ (αn × µ) for all n < ω because domϕ, domϕ′ are canonical summands of B. There- fore, [ϕ] = ⋃ n<ω ([ϕ] ∩ (αn × µ)) = ⋃ n<ω ([ ϕ′ ] ∩ (αn × µ) ) = [ ϕ′ ] . Secondly assume ρ = λ. Then (αn, εn) ∈ [ϕ] implies (εn, αn) ∈ [ϕ] and so (αn, αn) ∈ [ϕ] (see Definition 1.1.1). Similarly, we obtain (αn, αn) ∈ [ϕ′] and so, as in the first case, [ϕ] = ⋃ n<ω ([ϕ] ∩ (αn × αn)) = ⋃ n<ω ([ϕ] ∩ (µ × µ)) hαn,αn = = ⋃ n<ω ([ ϕ′ ] ∩ (µ × µ) ) hαn,αn = ⋃ n<ω ([ ϕ′ ] ∩ (αn × αn) ) = [ ϕ′ ] . In either case we deduce ϕ = ϕ′ and thus (2) is proven. It remains to show (3). So let ψ : B → B̂ be an R-homomorphism and let I ⊆ λ×ρ with \|I\| ≤ κ. By Proposition 1.1.5 there is a type τ ∈ T such that: ∃ϕ0 ∈ K ∀β0 ≥ ‖ϕ0‖ . . .∃ϕn ∈ K ∀βn ≥ ‖ϕn‖ . . . with (ϕn, βn)n<ω is τ -admissible where K = Kψ,I = {ϕ ∈ C\|ϕ ⊆ ψ, I ⊆ [ϕ]}. Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 17 We define a subset C of λ as follows: An ordinal α belongs to C if and only if α ≥ µ, α ≥ ‖ϕ0‖, Bαψ ⊆ B̂α (recall: Bα = ⊕ (β,ε)∈Tα Reβ,ε, Tα = α× (α∩ρ)), and, if (ϕ0, β0, . . . , ϕn, βn) is a finite part of one of the above τ -admissible sequences with βn < α, then there is also ϕn+1 ∈ K with [ϕn+1] ⊆ α × α and (ϕ0, β0, . . . , ϕn, βn, ϕn+1) is τ -admissible. Clearly, C is a cub and therefore the set E′ τ = {α ∈ Eτ \|Imηα ⊆ C} is stationary by Lemma 1.1.6. In the following let α ∈ E′ τ be fixed, i.e. ηα(n) ∈ C for all n < ω. By the definition of C we have ‖ϕ0‖ ≤ ηα(0) < ηα(1) and so there is ϕ1 ∈ K with ‖ϕ1‖ ≤ ηα(1) such that (ϕ0, ηα(0), ϕ1) is τ - admissible. We proceed like this along n < ω, i.e. whenever we have the τ -admissible sequence (ϕ0, ηα(0), . . . , ϕn, ηα(n)) with ‖ϕn‖ ≤ ηα(n) < ηα(n + 1) we can find ϕn+1 ∈ K with ‖ϕn+1‖ ≤ ηα(n + 1) such that (ϕ0, ηα(0), . . . , ϕn, ηα(n), ϕn+1) is τ -admissible. Therefore we obtain an infinite τ -admissible sequence (ϕn, ηα(n))n<ω, i.e. ‖ϕn‖ ≤ ηα(n) < ‖ϕn+1‖ and [ϕn+1] ∩ (ηα(n) × ηα(n)) = [ϕn] (cf. Definition 1.1.4). We put ϕ = ⋃ n<ω ϕn; then ‖ϕ‖ = sup n<ω ‖ϕn‖ = sup n<ω ηα(n) = α and [ϕ] ∩ (ηα(n) × ηα(n)) = ⋃ k≥n ([ϕk] ∩ (ηα(n) × ηα(n))) = [ϕn] . Hence ϕ = ⋃ n<ω ϕn ∈ Cα ⊆ C∗. Since α ∈ E′ τ was arbitrary and E′ τ is stationary the proof is finished. We finish this subsection with an “enumerated” version of the Strong Black Box Theorem 1.1.2, which can then directly be applied in Subsec- tion 1.2. Corollary 1.1.7. Let the assumptions be the same as in the Strong Black Box Theorem 1.1.2. Then there exists a family (ϕβ)β<λ of canonical homomorphism such that: (i) ‖ϕβ‖ ∈ E for all β < λ; (ii) ‖ϕγ‖ ≤ ‖ϕβ‖ for all γ ≤ β < λ; (iii) ‖ [ϕγ ]λ ∩ [ϕβ ] λ ‖ < ‖ϕβ‖ for all γ < β < λ; Jo u rn al A lg eb ra D is cr et e M at h .18 Strong Black Box (iv) Prediction: For any homomorphism ψ : B → B̂ and for any subset I of λ × ρ with \|I\| ≤ κ the set {α ∈ E\|∃β < λ : ‖ϕβ‖ = α, ϕβ ⊆ ψ, I ⊆ [ϕβ ]} is stationary. Proof. By the Strong Black Box Theorem 1.1.2 there is a class C∗ of canonical homomorphisms satisfying the conditions (i) and (iv) which are obviously independent of the enumeration (cf. conditions (1) and (3) in Theorem 1.1.2). Moreover, we put an arbitrary well-ordering on the sets Cα = {ϕ ∈ C∗\|‖ϕ‖ = α} (α ∈ E) and define ϕ ∈ Cα to be less than ϕ′ ∈ Cα′ if α < α′. This defines a well-ordering on C∗ and hence there is a corresponding ordinal λ∗ such that the condition (ii) is satisfied. In fact, λ∗ = λ since \|Cα\| ≤ µ for all α < λ and thus all initial segments of the above defined well-ordering are of cardinality less than λ. Condition (iii) also easily follows since ‖ [ϕ]λ∩[ϕ′]λ ‖ < ‖ϕ′‖ is obvious for ‖ϕ‖ < ‖ϕ′‖ and it coincides with condition (2) in Theorem 1.1.2 for ‖ϕ‖ = ‖ϕ′‖. 1.2. The Realization Theorem In this subsection we shall apply the Strong Black Box as given in Corol- lary 1.1.7 to prove the following theorem. Theorem 1.2.1. Let R, S, A and κ, µ, λ be as before. Then there exists an S-cotorsion-free R-module G of cardinality λ such that EndRG = A. Before we can construct the desired module we need the following lemma, which basically tells us how to obtain the module “step by step”. Step Lemma 1.2.2. Let P = ⊕ (α,ε)∈I∗ Reα,ε for some I∗ ⊆ λ × ρ and let M be an A-module as well as an S-cotorsion-free R-module with P ⊆∗ M ⊆∗ B̂. Also suppose that there is a set I = {(αn, εn)\|n < ω} ⊆ [P ] = I∗ such that α0 < α1 < . . . < αn < . . . and Iλ ∩ [g]λ is finite for all g ∈ M (Iλ = [I]λ = {αn\|n < ω}). Moreover, let ϕ : P → M be such an R-homomorphism which is not multiplication by an element of A. Then there exists an element y of P̂ such that yϕ /∈ M ′ := 〈M, yA〉∗ where ‘∗’ denotes the (S-) purification in B̂ and ϕ is identified with its unique extension ϕ : P̂ → M̂ . Moreover, M ′ is again an A-module as well as a S-cotorsion-free R- module with M ⊆∗ M ′ ⊆∗ B̂. Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 19 (The element y can be chosen to be either y = x or y = x + πb for suitable π ∈ R̂, b ∈ P and for x = ∑ n<ω qneαn,εn .) Proof. Let the assumptions be as above. Either x = ∑ n<ω qneαn,εn sat- isfies xϕ /∈ 〈M, xA〉∗ or not. In the latter case there are k < ω and a ∈ A such that qkxϕ − xa ∈ M. (+) Since M ⊆∗ B̂ is S-torsion-free and ϕ /∈ A we also have that qkϕ /∈ A, and thus there is an element b of P such that qkbϕ = b(qkϕ) 6= ba. Hence, by the cotorsion-freeness of M , there is π ∈ R̂ such that π(qkbϕ − ba) /∈ M. (++) Let z = x + πb and suppose zϕ ∈ 〈M, zA〉∗. Then qlzϕ − za′ ∈ M for some l ≥ k, a′ ∈ A. Therefore, using (+), we obtain that (qlzϕ − za′) − ql qk (qkxϕ − xa) = = qlxϕ + qlπbϕ − xa′ − πba′ − qlxϕ + ql qk xa = = x( ql qk a − a′) + π(qlbϕ − ba′) is an element of M . Now, [x]λ = Iλ while [ql(bϕ − ba′)]λ ∩ Iλ and[ x( ql qk a − a′) + π(qlbϕ − ba′) ] λ ∩ Iλ are both finite. Hence ql qk a − a′ = 0 and thus it follows from the above that π(qlbϕ− ql qk ba) = ql qk π(qkbϕ−ba) ∈ M ⊆∗ B̂. Since ql qk ∈ S this implies π(qkbϕ−ba) ∈ M contradicting (++). Therefore either y = x or y = z satisfies yϕ /∈ 〈M, yA〉∗ =: M ′. Clearly, M ′ is also an A-module. It remains to show that M ′ is S-cotorsion-free. Since M ′ ⊆∗ B̂ it is torsion-free and reduced. To prove Hom(R̂, M ′) = 0 let ϕ : R̂ → M ′ be a homomorphism and let k < ω such that qk(1ϕ) ∈ M + yA, say qk(1ϕ) = m + ya(m ∈ M, a ∈ A). Moreover, for any r ∈ R̂, let k ≤ kr < ω, mr ∈ M, ar ∈ A such that qkr (rϕ) = mr + yar. By the continuity of ϕ we have rϕ = r(1ϕ) and thus we deduce 0 = qkr (rϕ) − qkr r(1ϕ) = mr + yar − qkr qk r(m + ya), Jo u rn al A lg eb ra D is cr et e M at h .20 Strong Black Box respectively, mr − qkr qk rm = y( qkr qk ra − ar). Therefore, since [ mr − qkr qk rm ] λ ∩ Iλ is finite and [y]λ ∩ Iλ is infinite for either y = x or y = x + πb, we conclude that both sides of the above equation equal zero, i.e. ar = qkr qk ra and mr = qkr qk rm for each r ∈ R̂. Hence, by the purity of A in Â and of M in B̂ and since qkr qk ∈ S, we have that ra ∈ A and rm ∈ M for all r ∈ R̂, which implies a = 0 and m = 0 by the cotorsion-freeness of A and M . Thus 1ϕ = 0 and so ϕ is the zero-homomorphism as required. We are now ready to construct the desired module. Construction 1.2.3. Let (ϕβ)β<λ be a family of canonical homomor- phisms as given by Corollary 1.1.7. For any β < λ let Pβ = domϕβ, i.e. ϕβ : Pβ → P̂β . We inductively define elements yγ ∈ P̂γ and pure R-submodules Gβ of B̂ such that, for all γ < β < λ, (1) ‖yγ‖ = ‖Pγ‖(= ‖ϕγ‖), (2) Gβ = 〈B′, yγA(γ < β)〉 ∗ , and (3) Gβ is S-cotorsion-free. Recall that B′ = ⊕ α<λ eαA ⊇ B (see beginning of Subsection 1.1). Also note that the Gβs are then clearly A-modules. Let G0 = B′ ⊆∗ B̂ = B̂′; obviously B′ is S-cotorsion-free since A is, by assumption, and it also satisfies the conditions (1) and (2) since there are no relevant yγs. Next let β be a limit ordinal and suppose that Gγ satisfies all the required conditions for any γ < β. We put Gβ = ⋃ γ<β Gγ . Then Gβ certainly satisfies (1) and (2). Moreover, Gβ is clearly torsion-free and re- duced and so it remains to show Hom(R̂, Gβ) = 0. So, let ϕ : R̂ → Gβ be a homomorphism and let δ < β such that 1ϕ ∈ Gδ = 〈B′, yγA(γ < δ)〉 ∗ . Then, for each r ∈ R̂, we have [rϕ] ⊆ [1ϕ] and hence ‖ [rϕ]λ ∩ [yγ ]λ ‖ < ‖yγ‖ for all γ ≥ δ (see Corollary 1.1.7(iii)). Therefore rϕ ∈ Gδ for all r ∈ R̂, respectively, Imϕ ⊆ Gδ and thus ϕ = 0, i.e. Gβ is S-cotorsion-free. It remains to tackle the successor case. Assume Gβ is given satisfying all the conditions. We consider ϕβ . Since ‖ϕβ‖ ∈ E ⊆ λo there are (αn, εn) ∈ [ϕβ ] (n < ω) depending on β, such that α0 < α1 < . . . < αn < . . . and ‖ϕβ‖ = Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 21 supn<ω αn. We put I = {(αn, εn)\|n < ω}. Then ‖Iλ ∩ [g]λ ‖ < ‖ϕβ‖, respectively Iλ∩ [g]λ is finite, for all g ∈ Gβ by (1), (2) and condition (iii) in Corollary 1.1.7. We differentiate two cases. If ϕβ : Pβ → P̂β satisfies Imϕβ ⊆ Gβ and ϕβ /∈ A, then we apply the Step Lemma 1.2.2 to I as above, P = Pβ ⊆∗ B ⊆∗ Gβ and M = Gβ ⊆∗ B̂. We deduce the existence of an element y = yβ ∈ P̂β and of an A-module Gβ+1 = 〈Gβ , yβA〉∗ = 〈 B′, yγA(γ ≤ β) 〉 ∗ which is a S-cotorsion-free pure submodule of B̂ such that yβϕβ /∈ Gβ+1, where yβ = ∑ n<ω qneαn,εn or yβ = ∑ n<ω qneαn,εn + πb (π ∈ R̂, b ∈ B). Hence yβ satisfies (1) and Gβ+1 satisfies (2) and (3). If Imϕβ * Gβ or ϕβ ∈ A, then we put yβ = ∑ n<ω qneαn,εn and Gβ+1 = 〈Gβ , yβA〉∗. Then, also in this case, yβ and Gβ+1 satisfy all the required conditions (cf. Step Lemma 1.2.2). Finally, we define G by G = ⋃ β<λ Gβ = 〈B′, yβA(β < λ)〉∗. It is an immediate consequence from the construction that G is an A- module of cardinality λ which is also an S-cotorsion-free pure submodule of B̂. Next we describe the elements of G. Lemma 1.2.4. Let G be as in Construction 1.2.3. (a) The set {eα\|α < λ} ∪ {yβ \|β < λ} is linearly independent over A, i.e. 〈B′, yβA(β < λ)〉 = B′ ⊕ ⊕ β<λ yβA is a free A-module. (b) If g ∈ G \ B′ then there are a finite non-empty subset N of λ and k < ω such that qkg ∈ B′⊕ ⊕ β∈N yβA and [g]λ∩ [yβ ] λ is infinite iff β ∈ N . In particular, if ‖g‖ is a limit ordinal then ‖g‖ = ‖ymax N‖. Proof. First we show (a). We already know that {eα\|α < λ} is a linearly independent set since B′ = ⊕ α<λ eαA is a free A-module by definition. Now, it follows from Corollary 1.1.7(iii) that ‖ [yγ ]λ ∩ [yβ ] λ ‖ < ‖yβ‖, respectively that [yγ ]λ ∩ [yβ ] λ is finite, for γ < β < λ since [yβ ] λ ⊆ [ϕβ ] λ and ‖yβ‖ = ‖ϕβ‖ for all β. Moreover, [b]λ is finite for all b ∈ B′. Therefore the independence follows from the S-torsion-freeness of A together with Jo u rn al A lg eb ra D is cr et e M at h .22 Strong Black Box the fact that yβ ¹ eαn = qn for all but finitely many n < ω and for certain αn < λ, where g ¹ eα = gα for g = (eα, gα)α<λ ∈ B̂ = B̂′ ⊆∏ α<λ eαÂ(gα ∈ Â). It remains to show (b). So let g ∈ G\B′. Since G = 〈B′, yβA(β < λ)〉∗ there is k < ω such that qkg ∈ 〈B′, yβA(β < λ)〉 = B′ ⊕ ⊕ β<λ yβA. Therefore qkg = b + ∑ β∈N yβaβ (b ∈ B′, 0 6= aβ ∈ A, ∅ 6= N ⊆ λ finite) is a unique expression (for fixed k); in fact, for a k′ 6= k the expression only differs by an S-multiple, i.e. N is unique. Thus the conclusion follows from Corollary 1.1.7(iii) since [yβ ] λ ∩ [ yβ′ ] λ is finite for β 6= β′ and [qkg]λ = [g]λ. Using the above lemma we prove further properties of G. Lemma 1.2.5. Let G be as in Construction 1.2.3 and define Gα (α < λ) by Gα := {g ∈ G\|‖g‖ < α, ‖g‖A < α}. Then: (a) G ∩ P̂β ⊆ Gβ+1 for all β < λ; (b) {Gα\|α < λ} is a λ-filtration of G; and (c) if β < λ, α < λ are ordinals such that ‖ϕβ‖ = α then Gα ⊆ Gβ. Note, we used the upper index (β < λ) for the construction while we use the lower index (α < λ) for the filtration. Proof. First we show (a). Let g ∈ G ∩ P̂β for some β < λ. Since G0 = B′ ⊆ Gβ+1 we assume g ∈ G \ B′. Then, by Lemma 1.2.4, qkg ∈ B′ ⊕⊕ γ∈N yγA for some finite N ⊆ λ, k < ω such that [g]λ ∩ [yγ ]λ is infinite for γ ∈ N . Since g ∈ P̂β we also have [g]λ ⊆ [Pβ ] λ ( = [ P̂β ] λ ) . If ‖g‖ < ‖Pβ‖ then N ⊆ β by Corollary 1.1.7(ii) and thus g ∈ Gβ ⊆ Gβ+1. Otherwise, if ‖g‖ = ‖Pβ‖(∈ λo) then ‖g‖ = ‖yγ∗‖ = ‖ϕγ∗‖ for γ∗ = max N and [g]λ ∩ [yγ∗ ]λ ⊆ [ϕβ ] λ ∩ [ϕγ∗ ]λ is infinite. Hence β = γ∗ by condition (iii) of Corollary 1.1.7 and so g ∈ Gβ+1 as required. Condition (b) is obvious. To see (c) let β < λ, α < λ with ‖ϕβ‖ = α and let g ∈ Gα. If g ∈ B′ we are finished. Otherwise, by Lemma 1.2.4, we have qkg ∈ B′ ⊕ ⊕ γ∈N yγA (N ⊆ λ finite, k < ω) with [g]λ ∩ [yγ ]λ is infinite for γ ∈ N . This implies ‖ϕγ‖ = ‖yγ‖ ≤ ‖g‖ < α = ‖ϕβ‖ for all γ ∈ N and thus N ⊆ β by Corollary 1.1.7(ii), i.e. g ∈ Gβ , which finishes the proof. Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 23 Finally, we are ready to prove the main theorem of this subsection, i.e. the realization theorem. Proof of Theorem 1.2.1. Let G be the A-module as constructed in 1.2.3. We already know that G is an S-cotorsion-free R-module of cardinality λ. It remains to show EndRG = A. Obviously, A ⊆ EndRG. Conversely, suppose there exists ψ ∈ EndRG \ A. Let ψ′ = ψ ¹ B, then ψ′ /∈ A since ψ is uniquely deter- mined by ψ′ (B ⊆∗ G ⊆∗ B̂). Let I = {(αn, εn)\|n < ω} ⊆ λ×ρ such that α0 < α1 < . . . < αn < . . . and Iλ ∩ [g]λ is finite for all g ∈ G. Note, the existence of I can be easily arranged, e.g. let E Ã λo, α ∈ λo \ E, εn ∈ ρ (n < ω) arbitrary and (αn)n<ω any ladder on α. By the Step Lemma 1.2.2 there exists an element y of B̂ such that yψ /∈ 〈G, yA〉∗ = G′. By the Strong Black Box (Corollary 1.1.7) the set E′ = {α ∈ E\|∃β < λ : ‖ϕβ‖ = α, ϕβ ⊆ ψ′ ⊆ ψ, [y] ⊆ [ϕβ]} is stationary since \|[y]\| ≤ ℵ0 ≤ κ. Note, [y] ⊆ [ϕβ] implies y ∈ P̂β . Moreover, let C = {α < λ\|Gαψ ⊆ Gα}. Then C is a cub since {Gα\|α < λ} is a λ-filtration of G by Lemma 1.2.5(b). Now let α ∈ E′ ∩ C (6= ∅). Then Gαψ ⊆ Gα and there exists an ordinal β < λ such that ‖ϕβ‖ = α, ϕβ ⊆ ψ and y ∈ P̂β . The first property implies Gα ⊆ Gβ by Lemma 1.2.5(c) and the latter properties imply ϕβ /∈ A. Moreover, Pβ ⊆ B with ‖Pβ‖A ≤ ‖Pβ‖ = α and hence Pβ , and so also (Pβ)ψ are contained in Gα ⊆ Gβ . Therefore ϕβ : Pβ → Gβ with ϕβ /∈ A and thus it follows from the Construction 1.2.3 that yβϕβ /∈ Gβ+1. On the other hand, it follows from Lemma 1.2.5(a) that yβϕβ = yβψ ∈ G ∩ P̂β ⊆ Gβ+1 – a contradiction. So we have shown that no such ψ exists and this means EndRG = A as required. We would like to mention that one can also show, using standard arguments, that G is an ℵ1-free A-module. We finish this section with pointing out that the constructions and proofs in this section can be simplified for \|A\| ≤ κ. In this case we may work directly with B = ⊕ α<λ eαA and with P = ⊕ α∈I eαA as canonical summand provided \|I\| ≤ κ and (I ∩ µ)hα = I ∩ Imhα (α ∈ I) (cf. Definition 1.1.1). The definition of the equivalence relation on the set C of all canonical homomorphisms has to be adjusted: ϕ, ϕ′ are of the same type if [ϕ] ∩ µ = [ϕ′] ∩ µ and there is an order-isomorphism f : [ϕ] → [ϕ′] such that (eαa)ϕf̄ = (eαa)f̄ϕ′ = (eαfa)ϕ′ for all a ∈ A, α ∈ [ϕ] Jo u rn al A lg eb ra D is cr et e M at h .24 Strong Black Box (see Definition 1.1.3). All other adjustments are obvious (see also §2 for comparison). Note, the simplifications we can achieve in this way are due to the fact that the support function maps into λ rather than into λ × ρ. In fact, for \|A\| ≤ κ, we only need to assume that A is S-cotorsion-free, i.e. no “ρ”, respectively “F ”, is needed here (see beginning of §1). §2. Existence of E(R)-Algebras Throughout this second section let, as before, R be a commutative ring with 1 and let S be a countable multiplicatively closed subset of R con- taining no units except 1 such that R is (S-)cotorsion-free. We refer the reader to §1 for the definition of S-cotorsion-free. Here we additionally assume that R+ is torsion-free (as an abelian group). Also as before, we fix an enumeration S = {sn\|n < ω} of S with s0 = 1 and define a divisor chain (qn)n<ω by qn = s0 · . . . · sn to describe the S-adic topology of an R-module M by {qnM \|n < ω} as a basis of neighbourhoods of zero. In this section we show that, given infinite cardinals κ, µ, λ satisfying κ ≥ \|R\| , µκ = µ, λ = µ+, there exists an E(R)-algebra R̃ of cardinality λ which is also an S-cotorsion-free R-module. Recall, an R-algebra A is an E(R)-algebra if it satisfies EndR(AR) = A. The E(R)-algebra R̃ shall be constructed in Subsection 2.2 using a suitable, yet another, version of the Strong Black Box. This desired version will be introduced in the first subsection. For a proof of the same result using the General Black Box we refer to [DMV]. 2.1. The Black Box Theorem In this subsection we shall formulate the needed version of the Strong Black Box which will only be slightly different to the one given in Sub- section 1.1. Hence we shall only outline the proofs. Nevertheless, we will include all necessary definitions and results following the same pattern as in 1.1. Let R, S as well as κ, µ, λ be as above. As usual, we formulate the parameters of the Black Box with respect to a free R-module B and its S-adic completion B̂. In the present case, however, B is also a ring, namely a polynomial ring over R. Let B = R[Xα\|α < λ] be the polynomial ring in the commuting variables Xα and let M be the set of all monomials including the trivial monomial 1. Then B = ⊕ m∈M Rm. Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 25 For any g = (gmm)m∈M ∈ B̂ ⊆ ∏ m∈M R̂m we define the support of g by [g] = {m ∈ M\|gm 6= 0} and the support of M ⊆ B̂ by [M ] = ⋃ g∈M [g]; note \|[g]\| ≤ ℵ0 for all g ∈ B̂. Moreover, we define the X-support of g by [g]X = {α ∈ λ\|Xα occurs in some m ∈ [g]} ⊆ λ. Next we define a norm, as before, by ‖{α}‖ = α + 1 (α ∈ λ), ‖M‖ = supα∈M ‖{α}‖ (M ⊆ λ) and ‖g‖ = ‖ [g]X ‖ (g ∈ B̂), i.e. ‖g‖ = min{β ∈ λ\| [g]X ⊆ β}. Note, [g]X ⊆ β holds if and only if g is an element of B̂β where Bβ := R[Xα\|α < β]. As before, for a subset M of B̂ the above definitions extend naturally. Again, we need to say what we mean by a canonical homomorphism. For this we fix bijections hα : µ → α for all α with µ ≤ α < λ where we put hµ = idµ. For technical reasons we also put hα = hµ for α < µ. Definition 2.1.1. Let the bijections hα (α < λ) be as above. We define P to be a canonical subalgebra of B if P = R[Xα\|α ∈ I] for some I ⊆ λ with \|I\| ≤ κ such that (I ∩ µ) hα = I ∩ Imhα for all α ∈ I. Accordingly, an R-module homomorphism ϕ : P → B̂ is said to be a canonical homomorphism if P is a canonical subalgebra of B and Imϕ ⊆ P̂ ; we put [ϕ] = [P ], [ϕ]X = [P ]X and ‖ϕ‖ = ‖P‖. Let C denote the set of all canonical homomorphisms; clearly \|C\| = λ (as in §1). We are now ready to formulate the desired version of the Strong Black Box: Strong Black Box Theorem 2.1.2. Let κ, µ, λ be as before and let E ⊆ λo be a stationary subset of λ. Then there exists a family C∗ of canonical homomorphisms with the following properties: (1) If ϕ ∈ C∗ then ‖ϕ‖ ∈ E. (2) If ϕ, ϕ′ are two different elements of C∗ of the same norm α then ‖ [ϕ]X ∩ [ϕ′]X ‖ < α. (3) Prediction: For any R-homomorphism ψ : B → B̂ and for any subset I of λ with \|I\| ≤ κ the set {α ∈ E\|∃ϕ ∈ C∗ : ‖ϕ‖ = α, ϕ ⊆ ψ, I ⊆ [ϕ]X} is stationary. Jo u rn al A lg eb ra D is cr et e M at h .26 Strong Black Box Note that, although the above theorem reads exactly like the Strong Black Box Theorem in §1, the definition of a canonical homomorphism is slightly different to Definition 1.1.1. As mentioned before, we will not give all the details of the proof (again). However, we do state all used definitions and results, even when they coincide with their counterpart in §1. We begin by adjusting the definition of the equivalence relation on C: Definition 2.1.3. Canonical homomorphisms ϕ, ϕ′ are said to be equiv- alent, or of the same type (notation: ϕ ≡ ϕ′), if [ϕ]X ∩ µ = [ ϕ′ ] X ∩ µ and there exists an order-isomorphism f : [ϕ]X → [ϕ′]X such that (xf̄)ϕ′ = (xϕ)f̄ for all x ∈ domϕ where f̄ : d̂omϕ → d̂omϕ′ is the unique extension of the R-homomorphism defined by ( Xk0 α0 · · ·Xkn αn ) f̄ = Xk0 α0f · · ·X kn αnf (α0, . . . , αn ∈ [ϕ]X). Note, f : [ϕ]X → [ϕ′]X is unique since [ϕ]X , [ϕ′]X are well ordered. Thus, if ϕ ≡ ϕ′ and [ϕ]X = [ϕ′]X then f = id and so ϕ = ϕ′. As in §1 it is easy to see that there are at most µ different types (equivalence classes) in (C,≡). Next we recall the definition of an admissible sequence and of all other related notions: Definition 2.1.4. Let ϕ0 ⊂ ϕ1 ⊂ . . . ⊂ ϕn ⊂ . . . (n < ω) be an increas- ing sequence of canonical homomorphisms. Then (ϕn)n<ω is said to be admissible if [ϕ0]X ∩ µ = [ϕn]X ∩ µ and ‖ϕn‖ < ‖ϕn+1‖ for all n < ω. Also, we say that (ϕn)n<ω is admissible for a sequence (βn)n<ω of ordinals in λ (or (ϕn, βn)n<ω is admissible), if (ϕn)n<ω is admissible satisfying ‖ϕn‖ ≤ βn < ‖ϕn+1‖ and [ϕn]X = [ϕn+1]X ∩ βn for all n < ω. Moreover, two admissible sequences (ϕn)n<ω, (ϕ′ n)n<ω are said to be equivalent, or of the same type, if ϕn ≡ ϕ′ n for all n < ω. Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 27 Note, if (ϕn)n<ω is admissible then ϕ = ⋃ n<ω ϕn is an element of C with ‖ϕ‖ ∈ λo. Let T denote the set of all possible types of admissible sequences of canonical homomorphisms; clearly, \|T\| ≤ µℵ0 = µ. If (ϕn)n<ω, respectively (ϕn, βn)n<ω, is admissible of type τ , then we also use the notion τ -admissible. Moreover, if τ = (τn)n<ω ∈ T and (ϕn)n<k (k < ω) is a finite increasing sequence of canonical homomor- phisms satisfying ϕn ∈ τn and ‖ϕn‖ < ‖ϕn+1‖ for all n < k, then we shall also speak of (ϕn)n<k to be of type τ , keeping in mind that such a finite sequence could belong to different types in T. We are now ready to show the following result which is, as before, the “main ingredient” for the proof of the Strong Black Box Theorem 2.1.2. Because of this importance we do include a sketch of the proof. Proposition 2.1.5. Let ψ : B → B̂ be an R-homomorphism, I ⊆ λ a set of cardinality at most κ and K = Kψ,I = {ϕ ∈ C\|ϕ ⊆ ψ, I ⊆ [ϕ]X}. Then there exists a type τ ∈ T such that ∃ϕ0 ∈ K ∀β0 ≥ ‖ϕ0‖ . . .∃ϕn ∈ K ∀βn ≥ ‖ϕn‖ . . . with (ϕn, βn)n<ω being τ -admissible. Proof. Suppose, for contradiction, that the conclusion fails. Then, since the above formula is of “finite character”, we have for any type τ ∈ T, ∀ϕ0 ∈ K ∃β0(τ, ϕ0) ≥ ‖ϕ0‖ . . .∀ϕn ∈ K ∃βn(τ, ϕ0, . . . , ϕn) ≥ ‖ϕn‖ . . . with (ϕn, βn)n<ω not being τ -admissible. In the following we fix ordinals βn(τ, ϕ0, . . . , ϕn) as above (τ ∈ T, ϕi ∈ K, i ≤ n < ω). We define C to be the set of all α < λ such that Bαψ ⊆ B̂α (recall: Bα = R[Xβ \|β < α]) and βn(τ, ϕ0, . . . , ϕn) ≤ α for each type τ ∈ T and for any finite sequence (ϕ0, . . . , ϕn) of elements of K with ‖ϕi‖ ≤ α (iff [ϕi]X ⊆ α). Then C is an unbounded set (cf. proof of Proposition 1.1.5). Now we choose an increasing sequence α0 < α1 < . . . < αn < . . . in C with α0 ≥ µ, ‖I‖ and put α = supn<ω αn. Note that Bαψ ⊆ B̂α. Using these αns we inductively define subsets In of α = [Bα]X (n < ω) of cardinality at most κ by: I0 = I ∪ {αn\|n < ω} and In+1 = In ∪ In ψ ∪ In h where Jo u rn al A lg eb ra D is cr et e M at h .28 Strong Black Box In ψ = [(R[Xβ \|β ∈ In])ψ] X , In h = ⋃ β∈In ( (In ∩ µ)hβ ∪ (In ∩ Imhβ)h−1 β ) . We put I∗ = ⋃ n<ω In and P = R[Xβ \|β ∈ I∗]. It is easy to check that P is a canonical subalgebra satisfying ‖P‖ = α and Pψ ⊆ P̂ . Hence ϕ = ψ ¹ P is a canonical homomorphism with I ⊆ [ϕ]X , i.e. ϕ ∈ K. Finally, we put ϕn = ϕ ¹ (P ∩ Bαn). Using the same arguments as in the proof of Proposition 1.1.5 we deduce that (ϕn, βn)n<ω is a τ - admissible sequence for some type τ and for βn = βn(τ, ϕ0, . . . , ϕn). This contradiction finishes the proof. We have now provided all necessary definitions and results to prove the main theorem of this subsection. We also use Lemma 1.1.6 again, which has nothing to do with the special setting and hence it does not need to be adjusted. Proof of the Strong Black Box Theorem 2.1.2. Exactly as in the proof of Theorem 1.1.2, we decompose the given stationary set E into \|T\| ≤ µ pairwise disjoint stationary subsets, E = ⋃ τ∈T Eτ , and, for each τ ∈ T, we choose a ladder system {ηα\|α ∈ Eτ} such that the set {α ∈ Eτ \|Imηα ⊆ C} is stationary for any cub C (cf. Lemma 1.1.6). Also as in 1.1.2, we define C∗ = ⋃ α∈E Cα where, for each α ∈ Eτ , the set Cα consists of all canonical homomorphisms ϕ such that ‖ϕ‖ = α and ϕ = ⋃ n<ω ϕn for some τ -admissible sequence (ϕn)n<ω with [ϕn]X = [ϕ]X ∩ ηα(n) (n < ω). Note, for ϕ, ϕ′ ∈ Cα with domϕ = domϕ′ (iff [ϕ]X = [ϕ′]X) we deduce ϕ = ϕ′. Now, condition (1) is obviously satisfied. Condition (2) follows from [ϕ]X = ⋃ n<ω ([ϕ]X ∩ αn) = ⋃ n<ω ([ϕ]X ∩ µ) hαn = ⋃ n<ω ([ ϕ′ ] X ∩ µ ) hαn = ⋃ n<ω ([ ϕ′ ] X ∩ αn ) = [ ϕ′ ] X for µ ≤ αn ∈ [ϕ]X ∩ [ϕ′]X with supn<ω αn = ‖ϕ‖ = ‖ϕ′‖ (cf. 1.1.2). Finally, the proof of condition (3) is the same as the corresponding part of the proof of the Strong Black Box Theorem 1.1.2 using Proposi- tion 2.1.5 instead of Proposition 1.1.5. As in §1 we also present an “enumerated” version of the Strong Black Box Theorem. For the proof we refer to the proof of Corollary 1.1.7. Corollary 2.1.6. Let the assumptions be the same as in the Strong Black Box Theorem 2.1.2. Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 29 Then there exists a family (ϕβ)β<λ of canonical homomorphism such that (i) ‖ϕβ‖ ∈ E for all β < λ; (ii) ‖ϕγ‖ ≤ ‖ϕβ‖ for all γ ≤ β < λ; (iii) ‖ [ϕγ ]X ∩ [ϕβ ] X ‖ < ‖ϕβ‖ for all γ < β < λ; (iv) Prediction: For any R-homomorphism ψ : B → B̂ and for any subset I of λ with \|I\| ≤ κ the set {α ∈ E\|∃β < λ : ‖ϕβ‖ = α, ϕβ ⊆ ψ, I ⊆ [ϕβ] X } is stationary. 2.2. Constructing E(R)-algebras In this subsection we shall apply the Strong Black Box as given in Corol- lary 2.1.6 to prove the following theorem: Theorem 2.2.1. Let R, S and κ, µ, λ be as before. Then there exists an E(R)-algebra R̃ of cardinality λ which is also an S-cotorsion-free R-module. Before we construct the desired E(R)-algebra we need: Step Lemma 2.2.2. Let P = R[Xα\|α ∈ I∗] for some I∗ ⊆ λ and let M be an R-subalgebra of B̂ with P ⊆∗ M ⊆∗ B̂ which is an S-cotorsion-free R-module and a torsion-free abelian group. Also suppose that there is a set I = {α0 < α1 < . . . < αn < . . . (n < ω)} ⊆ I∗ = [P ]X such that I ∩ [g]X is finite for all g ∈ M . Moreover, let ϕ : P → M be such an R-homomorphism which is not multiplication by an element of M . Then there exists an element y of P̂ such that yϕ /∈ M ′ = (M [y])∗ where ‘∗’ denotes the S-purification in B̂, M [y] denotes the R-subalgebra of B̂ generated by M, y and ϕ is identified with its unique extension ϕ : P̂ → M̂ . Moreover, M ′ is again an R-subalgebra of B̂ which is also an S- cotorsion-free R-module with M ⊆∗ M ′ ⊆∗ B̂ and a torsion-free abelian group. (The element y can be chosen to be either y = x or y = x + πb for some suitable π ∈ R̂, b ∈ P and for x = ∑ n<ω qnXαn .) Note, it is straightforward that the purification of an R-subalgebra of B̂ is also an R-subalgebra. Jo u rn al A lg eb ra D is cr et e M at h .30 Strong Black Box Proof. Let the assumptions be as above. Either x = ∑ n<ω qnXαn sat- isfies xϕ /∈ (M [x])∗ or not. In the latter case there are k, n < ω, ri ∈ M(i ≤ n) such that qkxϕ = ∑ i≤n rix i. (+) Note, since ϕ is not multiplication by an element of M , also qkϕ /∈ M since M ⊆∗ B̂ is S-torsion-free. We differentiate two cases. First assume n ≤ 1. By the above P (qkϕ − r1) 6= 0 and hence there exists an element b of P such that 0 6= b(qkϕ − r1) = qkbϕ − br1 ∈ M . By the cotorsion-freeness of M there is π ∈ R̂ with π(qkbϕ − br1) /∈ M. (++) Let z = x + πb and suppose zϕ ∈ (M [z])∗. Then there are n′ < ω, k ≤ l < ω, ti ∈ M(i ≤ n′) such that qlzϕ = ∑ i≤n′ tiz i. Using (+) we obtain that qlπbϕ = qlzϕ − qlxϕ = ∑ i≤n′ ti(x + πb)i − ql qk (r0 + r1x). Since [πb] ⊆ [b] , [qlπbϕ] ⊆ [bϕ] and {Xi αn \|n < ω} ⊆ [ xi ] , respectively[ xi ] X = I, we deduce n′ = 1 and t1 = ql qk r1 by the assumption on I. Therefore qlπbϕ = t0 − ql qk r0 + ql qk r1πb and so ql qk π(qkbϕ − r1b) = t0 − ql qk r0 ∈ M ⊆∗ B̂ ( ql qk ∈ S), respectively π(qkbϕ − r1b) ∈ M contradicting (++). Now suppose n > 1 in (+). We may assume that rn 6= 0 and so 0 6= nrn ∈ M by the torsion-freeness of M+. Thus there is π ∈ R̂ with πnrn /∈ M. (+++) Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 31 Let z = x + π (i.e. b = 1 ∈ R ⊆ P ⊆ M) and suppose qlzϕ = ∑ i≤n′ tiz i for some n′ < ω, k ≤ l < ω, ti ∈ M(i ≤ n′). Using (+) we obtain qlπϕ = qlzϕ − qlxϕ = ∑ i≤n′ tiz i − ql qk ∑ i≤n rix i. Comparing the supports again we deduce n′ = n, tn = ql qk rn and tn−1 + tnπn = ql qk rn−1 and so ql qk rnπn = ql qk rn−1 − tn−1 ∈ M ⊆∗ B̂, respectively rnπn ∈ M, contradicting (+ + +). Therefore, in both cases, either y = x or y = z satisfies yϕ /∈ M ′ = (M [y])∗. The remaining properties of M ′ can be shown using support argu- ments and the assumptions on M (cf. proof of Lemma 1.2.2). We are now ready to construct the desired E(R)-algebra. Construction 2.2.3. Let (ϕβ)β<λ be a family of canonical homomor- phisms as given by Corollary 2.1.6. For any β < λ let Pβ = domϕβ = R[Xα\|α ∈ [ϕβ] X ]. We inductively define elements yγ ∈ P̂γ and R-subalgebras Rβ of B̂ such that, for all γ < β < λ, (1) ‖yγ‖ = ‖Pγ‖ (= ‖ϕγ‖), (2) Rβ = (B[yγ \|γ < β]) ∗ , (3) Rβ is an S-cotorsion-free R-module, and (4) Rβ is torsion-free as abelian group. Let R0 = B = R[Xα\|α < λ]; clearly B satisfies (2) and also B =⊕ m∈M Rm ⊆∗ B̂ is an S-cotorsion-free R-module and a torsion-free abelian group since R is, by assumption. Note, condition (1) is not rele- vant in this case. Next let β be a limit ordinal and suppose that Rγ satisfies all the required conditions for any γ < β. We put Rβ = ⋃ γ<β Rγ . Then Rβ Jo u rn al A lg eb ra D is cr et e M at h .32 Strong Black Box certainly satisfies (1), (2) and (4). Moreover, it is easy to check that Rβ is S-cotorsion-free (cf. Construction 1.2.3). It remains to tackle the successor case. Suppose Rβ is given satisfying all the conditions. We consider ϕβ. Since ‖ϕβ‖ ∈ λo there are ordinals α0 < α1 < . . . < αn < . . . (n < ω) in [ϕβ ] X such that ‖ϕβ‖ = supn<ω αn. We put I = {αn\|n < ω}. Then ‖I ∩ [g]X ‖ < ‖ϕβ‖, respectively I ∩ [g]X is finite, for all g ∈ Rβ by (1), (2) and condition (iii) in Corollary 2.1.6. We differentiate two cases: If ϕβ : Pβ → P̂β satisfies Imϕβ ⊆ Rβ and ϕβ /∈ Rβ then we apply the Step Lemma 2.2.2 to I as above, P = Pβ and M = Rβ ⊆∗ B̂. We thus deduce the existence of an element y = yβ ∈ P̂β and of a pure S-cotorsion-free R-submodule Rβ+1 of B̂ with Rβ+1 = ( Rβ [yβ ] ) ∗ = (B[yγ \|γ ≤ β]) ∗ which is also an R-algebra such that yβϕβ /∈ Rβ+1 and Rβ+1 is a torsion-free abelian group, where yβ = ∑ n<ω qnXαn or yβ = ∑ n<ω qnXαn + πb (π ∈ R̂, b ∈ Pβ). Therefore yβ satisfies (1) and Rβ+1 satisfies (2), (3) and (4). If Imϕβ * Rβ or ϕβ ∈ Rβ then we put yβ = ∑ n<ω qnXαn and Rβ+1 = ( Rβ [yβ ] ) ∗ . Then, also in this case, yβ and Rβ+1 satisfy all the required conditions. Finally we define R̃ by R̃ = ⋃ β<λ Rβ = (B[yβ \|β < λ]) ∗ . It is an immediate consequence from the construction that R̃ is an R-subalgebra of cardinality λ which is also a pure S-cotorsion-free R- submodule of B̂ and a torsion-free abelian group. Next we describe the elements of R̃. Lemma 2.2.4. Let R̃ be as in Construction 2.2.3. (a) The set {yβ \|β < λ} is linearly independent over B. (b) If g ∈ R̃ \ B then there are a finite non-empty subset N of λ and k < ω such that qkg ∈ B[yβ \|β ∈ N ] and [g]X ∩ [yβ ] X is infinite iff β ∈ N . In particular, if ‖g‖ is a limit ordinal then ‖g‖ = ‖ymax N‖. Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 33 Proof. The conclusion of (a) follows from the facts that [yβ ] X is infinite for all β < λ and [yβ ] X ∩ [ yβ′ ] X is finite for all β 6= β′ (see Corollary 2.1.6(iii)). To see (b) let g ∈ R̃ \ B. Since R̃ = (B[yβ \|β < λ]) ∗ there is k < ω such that qkg ∈ B[yβ \|β < λ] and so qkg ∈ B[yβ \|β ∈ N ] for some minimal subset N ⊂ λ where N 6= ∅ is obvious. Using the independence of the yβs the desired conditions are easily checked (cf. proof of Lemma 1.2.4). Using the above lemma we prove further properties of R̃. Lemma 2.2.5. Let R̃ be as in Construction 2.2.3 and define Rα (α < λ) by Rα := {g ∈ R̃\|‖g‖ < α}. Then: (a) R̃ ∩ P̂β ⊆ Rβ+1 for all β < λ; (b) Rα is an R-subalgebra of R̃ for all α < λ; (c) {Rα\|α < λ} is a λ-filtration of R̃; and (d) if β < λ, α < λ are ordinals such that ‖ϕβ‖ = α then Rα ⊆ Rβ. Note, we used the upper index (β < λ) for the construction while we use the lower index (α < λ) for the filtration. Proof. The proof of (a), (c) and (d) is similar to the one of Lemma 1.2.5 using Lemma 2.2.4 instead of Lemma 1.2.4. Condition (b) follows from Rα = (B′ α[yβ \|‖ϕβ‖ < α]) ∗ where B′ α = R[Xβ \|β + 1 < α]. Finally, we are ready to prove the main theorem of this subsection, i.e. the existence of an E(R)-algebra. Proof of Theorem 2.2.1. Let R̃ be the R-algebra as constructed in 2.2.3. We already know that R̃ is an S-cotorsion-free R-module of cardi- nality λ and also that R̃+ is torsion-free. It remains to show EndR(R̃R) = R̃. Clearly, R̃ ⊆ EndR(R̃R). Conversely, suppose there exists an R- module homomorphism ψ : R̃ → R̃ which is not multiplication by an element of R̃. Let ψ′ = ψ ¹ B. Then ψ′ /∈ R̃ since ψ is uniquely determined by ψ′ (B ⊆∗ R̃ ⊆∗ B̂). Let I = {α0 < α1 < . . . < αn < . . . (n < ω)} ⊆ λ such that I ∩ [g]X is finite for all g ∈ R̃. Note, the existence of I can be easily arranged, e.g. let E Ã λo, α ∈ λo \ E and (αn)n<ω any ladder on α. By the Step Lemma 2.2.2 there exists an element y of B̂ such that yψ /∈ ( R̃[y] ) ∗ . By the Strong Black Box (Corollary 2.1.6) the set E′ = {α ∈ E\|∃β < λ : ‖ϕβ‖ = α, ϕβ ⊆ ψ′ ⊆ ψ, [y] ⊆ [ϕβ]} Jo u rn al A lg eb ra D is cr et e M at h .34 Strong Black Box is stationary since \|[y]\| ≤ ℵ0 ≤ κ. Note, [y] ⊆ [ϕβ ] implies y ∈ P̂β where Pβ = domϕβ . Moreover, let C = {α < λ\|Rαψ ⊆ Rα}. Then C is a cub since {Rα\|α < λ} is a λ-filtration of R̃ by Lemma 2.2.5(c). Now let α ∈ E′ ∩ C (6= ∅). Then Rαψ ⊆ Rα and there exists an ordinal β < λ such that ‖ϕβ‖ = α, ϕβ ⊆ ψ and y ∈ P̂β . The first property implies Rα ⊆ Rβ by Lemma 2.2.5(d) and the latter properties imply ϕβ /∈ R̃, especially ϕβ /∈ Rβ . Moreover, Pβ ⊆ B with ‖Pβ‖ = α and hence Pβ , and so also (Pβ)ψ are contained in Rα ⊆ Rβ . Therefore ϕβ : Pβ → Rβ with ϕβ /∈ Rβ and thus it follows from the Construction 2.2.3 that yβϕβ /∈ Rβ+1. On the other hand, it follows from Lemma 2.2.5(a) that yβϕβ = yβψ ∈ R̃ ∩ P̂β ⊆ Rβ+1 – a contradiction. So we have shown that no such ψ exists and this means EndR(R̃R) = R̃ as required. Note, that one could also show, using standard arguments, that R̃ is an ℵ1-free R-module. §3. Existence of ultra-cotorsion-free modules Throughout this last section let, again, R be a commutative ring with 1 and let S be a countable multiplicatively closed subset of R containing no units except 1 such that R is (S-)cotorsion-free, that is, R is torsion-free and reduced (with respect to S) and satisfies HomR(R̂, R) = 0 where R̂ denotes the S-adic completion of R. Note, cotorsion-freeness for an arbitrary R-module M is defined in the same way (with HomR(R̂, M) = 0). Moreover, we fix an enumeration S = {sn\|n < ω} of S with s0 = 1 and define a divisor chain (qn)n<ω by qn = s0 · . . . · sn to describe the S-adic topology of an R-module M by {qnM \|n < ω} as a basis of neighbourhoods of zero. In this section we show that, given infinite cardinals κ, µ, λ satisfying κ ≥ \|R\| , µκ = µ, λ = µ+, there exists an ultra-cotorsion-free R-module G of cardinality λ. We define an R-module M to be ultra-cotorsion-free if M is S-cotorsion-free and, for any submodule H of M , if M/H is S-cotorsion- free then H = M or \|H\| < \|M \|. In particular, if M is ultra-cotorsion-free then M has no non-trivial S-cotorsion-free epimorphic images of smaller cardinality. Note, ultra-cotorsion-free modules have been used in [GSW] to show the abundance of cotorsion theories (cotorsion pairs). The desired R-module G shall be constructed using a suitable, yet again different, version of the Strong Black Box which will be introduced in the first subsection. Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 35 3.1. The Black Box Theorem In this subsection we shall formulate and prove the needed version of the Strong Black Box in full detail. Comparing the here presented version with the one given in §1 should enable the reader to understand that the Strong Black Box can be formulated (and proven) in rather differ- ent settings provided that the cardinalities in question are bounded by µ (number of types), respectively by λ (e.g. number of canonical sum- mands). Now, let R, S as well as κ, µ, λ be as above. As usual, we formulate the parameters of the Black Box with respect to a free R-module B and its S-adic completion B̂. Let B = ⊕ α<λ Reα. For any g = (gαeα)α∈λ ∈ B̂ ⊆ ∏ α∈λ R̂eα we define the support of g by [g] = {α ∈ λ\|gα 6= 0} ⊆ λ and the support of a subset M of B̂ by [M ] = ⋃ g∈M [g]; note \|[g]\| ≤ ℵ0 for all g ∈ B̂. Moreover, we define a norm on λ, respectively on B̂, by ‖{α}‖ = α+1 (α ∈ λ), ‖M‖ = supα∈M ‖{α}‖ (M ⊆ λ) and ‖g‖ = ‖ [g] ‖ (g ∈ B̂), i.e. ‖g‖ = min{β ∈ λ\| [g] ⊆ β}. Note, [g] ⊆ β holds iff g ∈ B̂β for Bβ = ⊕ α<β Reα. As before, for a subset M of B̂ the above definitions extend naturally. We also need to define canonical summands and other “canonical ob- jects” which shall play a crucial role in the formulation and the proof of the Strong Black Box. Note, in the here presented version we basically want to predict kernels of homomorphisms (i.e. submodules, respectively their elements), not the homomorphisms themselves. As before, we fix bijections hα : µ → α for all α with µ ≤ α < λ where we put hµ = idµ. For technical reasons we also put hα = hµ for α < µ. Definition 3.1.1. Let the bijections hα (α < λ) be as above. We define P to be a canonical summand of B if P = ⊕ α∈I Reα for some I ⊆ λ with \|I\| ≤ κ such that (I ∩ µ)hα = I ∩ Imhα for all α ∈ I. We call (P, v) a canonical pair if P is a canonical summand of B and v is a pure element of P̂ . Moreover, an infinite sequence (vn)n<ω of pure elements of B̂ satisfy- ing ‖vn‖ < ‖vn+1‖ (n < ω) is said to be a Signac-branch. A pair (P, (vn)n<ω) is called a canonical Signac-pair if P is a canonical summand of B and (vn)n<ω is a Signac-branch such that: • vn ∈ P̂ for all n < ω; and Jo u rn al A lg eb ra D is cr et e M at h .36 Strong Black Box • ‖P‖ = supn<ω ‖vn‖. Let P denote the set of all canonical pairs and let C denote the set of all canonical Signac-pairs. Then it is easy to see that \|P\| = \|C\| = λ. For the application in 3.2 we also need to include further parameters, namely pr(B)× R̂, where pr(B) denotes the set of all pure elements of B. Note, an element g of B̂ is called pure if g = sg′ (s ∈ S, g′ ∈ B̂) implies s = 1. In order to formulate the Strong Black Box according to our needs we define “traps” as follows: Definition 3.1.2. Let (P, (vn)n<ω) be a canonical Signac-pair, b a pure element of B and π ∈ R̂. Then t = (P, (vn)n<ω, b, π) is said to be a trap if b ∈ P (especially ‖b‖ < ‖P‖). We put [t] = [P ] and ‖t‖ = ‖P‖. We are now ready to present the desired version of the Strong Black Box: Strong Black Box Theorem 3.1.3. Let κ, µ, λ be as before and let E ⊆ λo be a stationary subset of λ. Then there exists a family C∗ of traps t = (Pt, (vt,n)n<ω, bt, πt) with the following properties: (1) If t ∈ C∗ then ‖t‖ ∈ E. (2) If t, t′ are two different elements of C∗ with ‖t‖ = ‖t′‖ = α then ‖ [t] ∩ [t′] ‖ < α. (3) Prediction: For any set U of pure elements of B̂ of cardinality λ, for any pure element b of B and for any π ∈ R̂ the set {α ∈ E\|∃t ∈ C∗ : ‖t‖ = α, {vt,n\|n < ω} ⊆ U, b = bt, π = πt} is stationary. To prove the above theorem we need further definitions and other results. We begin with an equivalence relation on P. Definition 3.1.4. Canonical pairs (P, v), (P ′, v′) are said to be equiva- lent or of the same type (notation: (P, v) ≡ (P ′, v′)) if [P ] ∩ µ = [ P ′ ] ∩ µ and there exists an order-isomorphism f : [P ] → [P ′] such that vf̄ = v′ Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 37 where f̄ : P̂ → P̂ ′ is the unique extension of the R-homomorphism defined by eαf̄ = eαf (α ∈ [P ]). Note, f : [P ] → [P ′] is unique since [P ], [P ′] are well ordered. Thus, if (P, v) ≡ (P ′, v′) and [P ] = [P ′] then f = id and so (P, v) = (P ′, v′). Obviously, any type in (P,≡) can be represented by a subset V of µ of cardinality at most κ, an order-type of a set M of cardinality at most κ and a countable sequence (αn, rn)n<ω with αn ∈ M , rn ∈ R (which describes v). Therefore there are at most µκ ·κκ ·κℵ0 · \|R\|ℵ0 = µ different types in (P,≡). Next we consider certain infinite sequences of canonical pairs: Definition 3.1.5. A sequence (Pn, vn)n<ω of canonical pairs is said to be admissible if P0 ⊂ P1 ⊂ . . . ⊂ Pn ⊂ . . . (n < ω), [P0] ∩ µ = [Pn] ∩ µ and ‖Pn‖ < ‖vn+1‖(≤ ‖Pn+1‖) for each n < ω. Also, we say that (Pn, vn)n<ω is admissible for a sequence (βn)n<ω of ordinals in λ, if (Pn, vn)n<ω is admissible such that ‖Pn‖ ≤ βn < ‖Pn+1‖ and [Pn] = [Pn+1] ∩ βn for all n < ω. Moreover, two admissible sequences (Pn, vn)n<ω, (P ′ n, v′n)n<ω are said to be equivalent or of the same type, if (Pn, vn) ≡ (P ′ n, v′n) for all n < ω. Note, if (Pn, vn)n<ω is admissible then ( P = ⋃ n<ω Pn, (vn)n<ω ) is a canonical Signac-pair since ‖P‖ = supn<ω ‖Pn‖ = supn<ω ‖vn‖. Let T denote the set of all possible types of admissible sequences of canonical pairs. It follows immediately from the above definition that any type τ in T can be represented by τ = (τn)n<ω where the τns are equivalence classes of (P,≡) with the same underlying subset V of µ. Hence we deduce \|T\| ≤ µℵ0 = µ. If (Pn, vn)n<ω is an admissible sequence of type τ , then we also use the notion τ -admissible. Moreover, if τ = (τn)n<ω ∈ T and (Pn, vn)n<k (k < ω) is a finite increasing sequence of canonical pairs satisfying (Pn, vn) ∈ τn for all n < k, then we shall also speak of (Pn, vn)n<k to be of type τ , keeping in mind that such a finite sequence could belong to different types in T. Jo u rn al A lg eb ra D is cr et e M at h .38 Strong Black Box The next result is the key for proving the Strong Black Box Theo- rem 3.1.3. Note again, that the kind of formula we use to formulate this result goes under the generic name “Svenonius sentences” (cf. [H, p.112], also §1). Proposition 3.1.6. Let U be a set of pure elements of B̂ of cardinality λ, let b be a pure element of B, and let K = KU,b = {(P, v) ∈ P\|v ∈ U, b ∈ P}. Then there exists a type τ ∈ T such that ∃(P0, v0) ∈ K ∀β0 ≥ ‖P0‖ . . .∃(Pn, vn) ∈ K ∀βn ≥ ‖Pn‖ . . . with (Pn, vn)n<ω being τ -admissible for (βn)n<ω. Proof. Suppose, for contradiction, that the conclusion fails. Then, for any τ ∈ T, we have: ∀(P0, v0) ∈ K ∃β0(τ, P0, v0) ≥ ‖P0‖ . . . ∀(Pn, vn) ∈ K ∃βn(τ, P0, v0, . . . , Pn, vn) ≥ ‖Pn]‖ . . . with (Pn, vn)n<ω not being τ -admissible for (βn)n<ω. In the following we fix ordinals βn(τ, P0, v0, . . . , Pn, vn) as above (τ ∈ T, (Pi, vi) ∈ K, i ≤ n < ω). We define C to be the set of all α < λ such that βn(τ, P0, v0, . . . , Pn, vn) ≤ α for each τ ∈ T and for any finite sequence (Pi, vi)i≤n with (Pi, vi) ∈ K and ‖Pi‖ ≤ α (i ≤ n). Then C is unbounded since, starting with an arbitrary α0 < λ, one can inductively define ordinals αk (k < κ+ ≤ µ) such that α = sup{αk\|k < κ+} is an element of C (cf. proof of Proposition 1.1.5). Now, we inductively choose elements vn of U and ordinals αn in C such that α0 ≥ µ, ‖b‖ and ‖vn‖ ≤ αn < ‖vn+1‖ for all n < ω; this is possible since C is unbounded and \|U \| = λ, so the norms of the elements of U also form an unbounded set in λ. We put α = supn<ω αn = supn<ω ‖vn‖ and define subsets In of α of cardinality at most κ by: I0 = [b] ∪ {αn\|n < ω} ∪ ⋃ n<ω [vn] and In+1 = In ∪ ⋃ β∈In ( (In ∩ µ)hβ ∪ (In ∩ Imhβ)h−1 β ) . We put I∗ = ⋃ n<ω In. Then it is easy to check that P = ⊕ α∈I∗ Reα is a canonical summand of B such that b ∈ P , vn ∈ P̂ for all n < ω and ‖P‖ = ‖I∗‖ = α = supn<ω ‖vn‖, i.e. (P, (vn)n<ω) is a canonical Signac-pair. Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 39 Finally, let Pn = P ∩ Bαn (Bαn = ⊕ β<αn Reβ), i.e. [Pn] = [P ] ∩ αn. Then b ∈ P0 (⊆ Pn) since ‖b‖ ≤ α0 ([b] ⊆ α0) by the definition of C and vn ∈ P̂n for any n < ω since ‖vn‖ ≤ αn ([vn] ⊆ αn) by the choice of the vns and αns. Hence (Pn, vn) ∈ K for all n < ω. Moreover, [P0] ∩ µ = [Pn] ∩ µ = [P ] ∩ µ (n < ω) since µ ≤ α0 < α1 < . . . < αn < . . . (n < ω). Therefore (Pn, vn)n<ω is an admissible sequence, say of type τ ∈ T. By the definition of C we also have that ‖Pn‖ ≤ βn = βn(τ, P0, . . . , Pn) ≤ αn for any n < ω and thus [Pn+1]∩βn = [Pn]∩βn = [Pn]. Hence (Pn, vn)n<ω is τ -admissible for (βn)n<ω, contradicting the assumption that it is not for βn = βn(τ, P0, . . . , Pn). Therefore the original conclusion holds and so the proof is finished. We are now ready to prove the main theorem of this subsection. Proof of the Strong Black Box Theorem 3.1.3. Let pr(B) be the set of all pure elements of B and put P = pr(B)× R̂. Then \|P\| = λ ·2κ = λ. First we decompose the given stationary set E into \|T\| ≤ µ pairwise disjoint stationary subsets, say E = ⋃ τ∈T Eτ . Moreover, for each τ ∈ T, we decompose Eτ into \|P\| = λ pairwise disjoint stationary subsets: Eτ =⋃ p∈P Eτ,p. Note, for p = (b, π) ∈ P and τ ∈ T, we may assume that ‖b‖ < α for all α ∈ Eτ,p. For each τ ∈ T and each p ∈ P we choose a ladder system {ηα\|α ∈ Eτ,p} such that the set {α ∈ Eτ,p\|Imηα ⊆ C} is stationary for any cub C (cf. Lemma 1.1.6). Let τ ∈ T, p = (b, π) ∈ P and α ∈ Eτ,p; note ‖b‖ < α. We define Cα to be the set of all traps t = (P, (vn)n<ω, b, π) such that ‖t‖ = ‖P‖ = α and P = ⋃ n<ω Pn for some τ -admissible sequence (Pn, vn)n<ω of canonical pairs with [Pn] = [P ] ∩ ηα(n). Note, for t, t′ ∈ Cα with [t] = [t′] (iff P = P ′), we clearly deduce t = t′ since bt = b = bt′ , πt = π = πt′ , and [Pt,n] = [ Pt′,n ] , (Pt,n, vt,n) ≡ (Pt′,n, vt′,n) imply (Pt,n, vt,n) = (Pt′,n, vt′,n) for all n < ω (cf. Definitions 3.1.2 and 3.1.4). Now, we define C∗ to be the union of all these Cαs, i.e. C∗ = ⋃ α∈E Cα. Clearly, condition (1) is satisfied. To see (2) let t, t′ ∈ C∗ with ‖t‖ = ‖t′‖ = α. Then t, t′ ∈ Cα and thus t = (P, (vn)n<ω, b, π), t′ = (P ′, (v′n)n<ω, b′, π′) with b = b′, π = π′, and (P, (vn)n<ω), (P ′, (v′n)n<ω) are of the same type τ (α ∈ Eτ,(b,π)) where P = ⋃ n<ω Pn, P ′ = ⋃ n<ω P ′ n, [P ] ∩ ηα(n) = [Pn], [P ′ n] ∩ ηα(n) = [P ′ n]. Suppose, for contradiction, that ‖ [t] ∩ [t′] ‖ = α (recall: [t] = [P ] and [t′] = [P ′]). Then there are αn ∈ [P ] ∩ [P ′] with supn<ω αn = α (w.l.o.g. αn ≥ µ). Now, ([P ]∩µ)hαn = [P ]∩αn and ([P ′]∩µ)hαn = [P ′]∩αn since Jo u rn al A lg eb ra D is cr et e M at h .40 Strong Black Box P , P ′ are canonical summands of B (see Definition 3.1.1). Moreover, [P ] ∩ µ = [P0] ∩ µ = [ P ′ 0 ] ∩ µ = [ P ′ ] ∩ µ and therefore [P ] = ⋃ n<ω ([P ] ∩ αn) = ⋃ n<ω ([ P ′ ] ∩ αn ) = [ P ′ ] . This implies (P, (vn)n<ω) = (P ′, (v′n)n<ω) and so property (2) is proven. It remains to show (3). To do so let U be a set of pure elements of B̂ of cardinality λ and let p = (b, π) ∈ P = pr(B)× R̂. By Proposition 3.1.6 there is a type τ ∈ T such that ∃(P0, v0) ∈ K ∀β0 ≥ ‖P0‖ . . .∃(Pn, vn) ∈ K ∀βn ≥ ‖Pn‖ . . . with (Pn, vn)n<ω is τ -admissible for (βn)n<ω where K = {(P, v) ∈ P\|v ∈ U, b ∈ P}. Let C be the set of all ordinals α < λ such that α ≥ µ, α ≥ ‖P0‖ and, if (Pn, vn)n≤k is a finite part of one of the above τ -admissible sequences for (βn)n≤k with βk < α, then there is (Pk+1, vk+1) ∈ K with [Pn+1] ⊆ α and (Pn, vn)n≤k+1 is τ -admissible. Obviously, C is a cub. Therefore E′ τ,p = {α ∈ Eτ,p\|Imηα ⊆ C} is stationary. In the following let α ∈ E′ τ,p be fixed, i.e. ηα(n) ∈ C for all n < ω. By the definition of C we have ‖P0‖ ≤ ηα(0) < ηα(1) and so there is (P1, v1) ∈ K with ‖P1‖ ≤ ηα(1) such that (Pn, vn)n≤1 is τ -admissible for (ηα(0), ηα(1)). We proceed like this for each n < ω, i.e. whenever we have a sequence (Pn, vn)n≤k which is τ -admissible for (ηα(n))n≤k we can find (Pk+1, vk+1) ∈ K with ‖Pk+1‖ ≤ ηα(k + 1) such that (Pn, vn)n≤k+1 is τ - admissible for (ηα(n))n≤k+1. Therefore we obtain an infinite τ -admissible sequence (Pn, vn)n<ω with ‖Pn‖ ≤ ηα(n) and [Pn+1] ∩ ηα(n) = [Pn]. We put P = ⋃ n<ω Pn. Then ‖P‖ = supn<ω ‖Pn‖ = supn<ω ηα(n) = α and [P ] ∩ ηα(n) = ⋃ i≥n ([Pi] ∩ ηα(n)) = [Pn]. Hence (P, (vn)n<ω, b, π) ∈ Cα. Since α ∈ E′ τ,p was arbitrary and E′ τ,p is stationary, the proof is finished. We finish this subsection with an “enumerated” version of the Strong Black Box. For the proof we refer the reader to the proof of Corol- lary 1.1.7. Corollary 3.1.7. Let the assumptions be the same as in the Strong Black Box Theorem 3.1.3. Then there exists a family (tβ = (Pβ , (vβ,n)n<ω, bβ , πβ)) β<λ of traps such that Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 41 (i) ‖tβ‖ ∈ E for all β < λ; (ii) ‖tγ‖ ≤ ‖tβ‖ for all γ ≤ β < λ; (iii) ‖ [tγ ] ∩ [tβ] ‖ < ‖tβ‖ for all γ < β < λ; (iv) Prediction: For any set U of pure elements of B̂ of cardinality λ, for any pure element b of B and for any π ∈ R̂, the set {α ∈ E\|∃β < λ : ‖tβ‖ = α, {vβ,n\|n < ω} ⊆ U, b = bβ , π = πβ} is stationary. 3.2. Constructing ultra-cotorsion-free modules In this final subsection we shall apply the Strong Black Box as given in Corollary 3.1.3 to prove the following theorem: Theorem 3.2.1. Let R, S and κ, µ, λ be as before. Then there exists an ultra-cotorsion-free R-module G of cardinality λ. Before we construct the desired module we show: Step Lemma 3.2.2. Let M be a pure S-cotorsion-free submodule of B̂, let b be a pure element of B ∩ M and let π ∈ R̂. Moreover, let v = (vn)n<ω be a Signac-branch with vn ∈ M for all n < ω such that ‖b‖ < ‖v‖ (= supn<ω ‖vn‖) and ‖ [m] ∩ [v] ‖ < ‖v‖ for all m ∈ M . Then M ′ = 〈 M, y = ∑ n<ω qnvn + πb 〉 ∗ is also S-cotorsion-free. Proof. Let the assumptions be as above and consider a homomorphism ϕ : R̂ → M ′. Since 1ϕ ∈ M ′ there is k < ω such that qk(1ϕ) ∈ M + Ry, say qk(1ϕ) = m + ry for some m ∈ M, r ∈ R. Moreover, for any ρ ∈ R̂, let k ≤ kρ < ω, mρ ∈ M, rρ ∈ R such that qkρ (ρϕ) = mρ + rρy. Hence, since ρϕ = ρ(1ϕ) by the continuity of ϕ, we deduce 0 = qkρ (ρϕ) − qkρ ρ(1ϕ) = mρ + rρy − qkρ qk ρ(m + ry), respectively mρ − qkρ qk ρm = ( qkρ qk ρr − rρ)y =: g. Jo u rn al A lg eb ra D is cr et e M at h .42 Strong Black Box From g = mρ − qkρ qk ρm it follows that g ∈ R̂M and thus ‖ [g] ∩ [v] ‖ < ‖v‖ by the assumption. On the other hand, g = ( qkρ qk ρr − rρ)y and so ‖ [g] ∩ [v] ‖ = ‖v‖ = ‖y‖ unless rρ = qkρ qk ρr, i.e. g = 0. Therefore qkρ qk ρr = rρ ∈ R and qkρ qk ρm = mρ ∈ M for all ρ ∈ R̂ and so, since qkρ qk ∈ S, ρr ∈ R and ρm ∈ M for all ρ ∈ R̂. The cotorsion-freeness of R and M now implies r = 0, m = 0 and thus 1ϕ = 0, respectively ϕ = 0, as required. We are now ready to construct the desired module. Construction 3.2.3. Let (tβ = (Pβ , (vβ,n)n<ω, bβ , πβ)) β<λ be a family of traps as given by Corollary 3.1.7. We inductively define elements yγ ∈ P̂γ and pure submodules Gβ of B̂ such that, for all γ < β < λ, (1) yγ = 0 or ‖yγ‖ = ‖Pγ‖ (= ‖tγ‖), (2) Gβ = 〈B, yγ(γ < β)〉 ∗ , and (3) Gβ is S-cotorsion-free. First we put G0 = B = ⊕ α<λ Reα; B is clearly a pure S-cotorsion-free submodule of B̂ satisfying (2). Note, condition (1) is not relevant. Next let β be a limit ordinal and suppose the Gγs (γ < β) are given satisfying all the required conditions. We put Gβ = ⋃ γ<β Gγ . Then, obviously, the yγs and Gβ satisfy (1) and (2). Moreover, Gβ is S-cotorsion- free since, for a homomorphism ϕ : R̂ → Gβ , we have 1ϕ ∈ Gγ for some γ < β and [ρϕ] ⊆ [1ϕ] for all ρ ∈ R̂; thus we obtain Imϕ ⊆ Gγ by (1), (2) and condition (iii) in Corollary 3.1.7. Now, suppose that Gβ is given satisfying the above properties and let tβ = (Pβ , (vβ,n)n<ω, bβ , πβ) be the trap from the above family. We differentiate two cases. If vβ,n ∈ Gβ for all n < ω, then we define yβ ∈ P̂β by yβ = ∑ n<ω qnvβ,n + πβbβ and put Gβ+1 = 〈 Gβ, yβ 〉 ∗ = 〈B, yγ(γ ≤ β)〉 ∗ . From the Step Lemma 3.2.2 we know that Gβ+1 is also a pure S-cotorsion- free submodule of B̂. Moreover, yβ 6= 0 satisfies (1) since ‖yβ‖ = supn<ω ‖vβ,n‖ and Gβ+1 satisfies (2) and (3). Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 43 If vβ,n /∈ Gβ for some n < ω then we do not extend Gβ , i.e. we put Gβ+1 = Gβ (yβ = 0). Clearly, the above conditions remain satisfied. Finally, let G = ⋃ β<λ Gβ = 〈B, yβ(β < λ)〉∗. Note, that Gβ 6= Gβ+1 (i.e. yβ 6= 0) happens “often” since the predic- tion in Corollary 3.1.7 can, for example, be applied to the set U = pr(B) of all pure elements of B = G0. It is immediate from the construction that \|G\| = λ and that G is a pure S-cotorsion-free submodule of B̂. Next we describe the elements of G. Lemma 3.2.4. Let G be as in Construction 3.2.3. (a) The set {eα\|α < λ} ∪ {yβ \|β < λ, yβ 6= 0} is linearly independent (over R), i.e. 〈B, yβ(β < λ)〉 = B⊕ ⊕ β<λ Ryβ is a free R-module. (b) If g ∈ G\B then there are a finite non-empty subset N of λ and k < ω such that qkg ∈ B⊕ ⊕ β∈N Ryβ and ‖ [g]∩[yβ ] ‖ = ‖yβ‖ = ‖tβ‖ iff β ∈ N . In particular, if ‖g‖ is a limit ordinal then ‖g‖ = ‖ymax N‖. Proof. The conclusion of part (a) follows easily from ‖ [yγ ] ∩ [yβ ] ‖ < ‖yβ‖ = ‖tβ‖ for γ < β < λ and yβ 6= 0. Part (b) follows from qkg ∈ B ⊕ ⊕ β<λ Ryβ for some k < ω (cf. proofs of the Lemmas 1.2.4 and 2.2.4). Using the above lemma we prove further properties of the module G. Lemma 3.2.5. Let G be as in Construction 3.2.3 and define Gα (α < λ) by Gα := {g ∈ G\|‖g‖ < α}. Then: (a) {Gα\|α < λ} is a λ-filtration of G; (b) if β < λ, α < λ are ordinals such that ‖tβ‖ = α then Gα ⊆ Gβ; (c) if α /∈ E then Gα+1/Gα is free; and (d) if α ∈ E and Gα+1/Gα 6= 0 then Gα+1/Gα contains a non-zero S-divisible submodule. Proof. First we show (a). Let α < λ be arbitrary. Then we clearly have Gα ⊆ Gα+1. Moreover, \|Gα\| ≤ ∣∣∣B̂α ∣∣∣ ≤ \|Bα\| ℵ0 = (\|R\| · \|α\|)ℵ0 ≤ µℵ0 = µ < λ (recall: Bα = ⊕ δ<α Reδ). It is easy to see that the increasing chain of the Gαs is smooth and that G = ⋃ α<λ Gα holds. Jo u rn al A lg eb ra D is cr et e M at h .44 Strong Black Box To see (b) let β < λ, α < λ with ‖tβ‖ = α and let g ∈ Gα. If g ∈ B we are finished. Otherwise, by Lemma 3.2.4, qkg ∈ B ⊕ ⊕ γ∈N Ryγ for some finite N ⊆ λ and k < ω such that [g]∩ [yγ ] is infinite iff γ ∈ N . This implies ‖tγ‖ = ‖yγ‖ ≤ ‖g‖ < α = ‖tβ‖ for all γ ∈ N and thus N ⊆ β by Corollary 3.1.7(ii). Hence g ∈ Gβ ⊆∗ G and so (b) is proven. Next we show (c). Let α < λ with α /∈ E. If α is a limit ordinal then, by Corollary 3.1.7(i) and Lemma 3.2.4, there is no element of norm α in G and so Gα+1/Gα = 0 in this case. If α = δ + 1 (δ ∈ λ) then ‖eδ‖ = α and any element g ∈ G with ‖g‖ = α can be written as g = reδ + g′ (r ∈ R, g′ ∈ Gα). Therefore Gα+1/Gα = 〈eδ + Gα〉 ∼= R in this case. Thus Gα+1/Gα is free for α /∈ E as required. Finally we show (d). To do so let α ∈ E with Gα 6= Gα+1. Then there is an element g ∈ G with ‖g‖ = α. It hence follows from Lemma 3.2.4 that there exists β < λ such that yβ 6= 0 and α = ‖yβ‖ (= ‖tβ‖). This implies Gβ 6= Gβ+1 and hence vβ,n ∈ Gβ with ‖vβ,n‖ < α (see Construction 3.2.3 and Definition 3.1.1 of a Signac-branch). Therefore vβ,n ∈ Gα for all n < ω. We also know that bβ ∈ Pβ ⊆ Gα. Thus we deduce yβ ≡ qn (∑ k≥n qk qn vβ,k + π (n) β bβ ) mod Gα for all n < ω, where πβ − qnπ (n) β ∈ R. So yβ + Gα is an S-divisible element of Gα+1/Gα. Note, that not all of the above properties are of importance in the here considered context; we included them for completeness. Finally we are ready to prove the main theorem of this subsection, i.e. the existence of an ultra-cotorsion-free R-module G. Proof of Theorem 3.2.1. Let G be the R-module as constructed in 3.2.3. We already know that \|G\| = λ and that G ⊆∗ B̂ is S-cotorsion- free. It remains to show that G is ultra-cotorsion-free. To do so let H be a submodule of G and let ψ : G → G/H be the canonical epimorphism. Suppose, for contradiction, that 0 6= G/H is S-cotorsion-free and \|H\| = λ (= \|G\|). Since G/H is S-reduced and G/B ⊆∗ B̂/B is S-divisible H cannot contain B. Therefore, by the S-torsion-freeness of G/H (iff H ⊆∗ G), there is a pure element b of B such that b /∈ H(= kerψ), i.e. bψ 6= 0. We are going to show that 0 6= πbψ ∈ G/H for all π ∈ R̂. Let U = pr(H) and let π ∈ R̂ be arbitrary. Then \|U \| = λ and so, by Corollary 3.1.7(iv), there exists β < λ such that {vβ,n\|n < ω} ⊆ U ⊆ H ⊆ G, b = bβ and π = πβ. Let ‖Pβ‖ = α. By the definitions of a Signac- branch and of a trap we then obtain {vβ,n\|n < ω} ⊆ Gα where Gα ⊆ Gβ by Lemma 3.2.5. Therefore it follows from the Construction 3.2.3 that Jo u rn al A lg eb ra D is cr et e M at h .R. Göbel, S. L. Wallutis 45 0 6= yβ (= ∑ n<ω qnvβ,n + πb) ∈ G and so yβψ ∈ G/H. Identifying ψ with its unique extension to B̂ and using the continuity of ψ we deduce yβψ = ( ∑ n<ω qnvβ,n ) ψ + (πb)ψ = ∑ n<ω qn (vβ,nψ)︸︷︷︸ =0 +π(bψ) = π(bψ) and thus π(bψ) ∈ G/H. Since π ∈ R̂ was arbitrary and bψ 6= 0, we deduce 0 6= π(bψ) ∈ G/H for all π ∈ R̂, contradicting the S-cotorsion- freeness of G/H. Therefore such a submodule H does not exist, i.e. for G/H to be S-cotorsion-free we need G = H or \|H\| < λ as required, and so the proof is finished. Finally note, that the above R-module can be shown to be ℵ1-free using standard arguments (e.g. see [GSW] or [P]). References [C] A.L.S. Corner: Every countable reduced torsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 13 (1963), 687–710. [CG] A.L.S. Corner, R. Göbel: Prescribing endomorphism algebras – A unified treatment, Proc. London Math. Soc. (3) 50 (1985), 447–479. [DMV] M. Dugas, A. Mader, C. Vinsonhaler: Large E-rings exist, J. Algebra 108 (1987), 88–101. [EM] P.C. Eklof, A.H. Mekler: Almost free modules, North–Holland (1990). [GSW] R. Göbel, S. Shelah, S.L. Wallutis: On the lattice of cotorsion theories, J. Algebra 238 (2001), 292–313. [H] W. Hodges: Model Theory, Encyclopedia of Mathematics and its Applications 42, Cambridge University Press (1993). [J] T. Jech: Set Theory, Academic Press (1978). [P] S.L. Pabst (Wallutis): On ℵ1-free modules with trivial dual, Comm. Algebra 28 (2000), 5053–5065. [S] S. Shelah: A combinatorial theorem and endomorphism rings of abelian groups II, Abelian Groups and Modules, CISM Courses and Lectures 287 (1984), 37–86. Contact information R. Göbel FB 6 - Mathematik, Universität Duisburg- Essen, 45117 Essen, Germany E-Mail: r.goebel@uni-essen.de URL: http://www.uni-essen.de/goebel/ S.L. Wallutis FB 6 - Mathematik, Universität Duisburg- Essen, 45117 Essen, Germany E-Mail: simone.wallutis@uni-essen.de Received by the editors: 23.05.2003 and final form in 13.11.2003.

An algebraic version of the Strong Black Box

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