On quantales of preradical Bland filters and differential preradical filters

On quantales of preradical Bland filters and differential preradical filters

We prove that the set of all Bland preradical filters over an arbitrary differential ring form a quantale with respect to meets where the role of multiplication is played by the usual Gabriel pro-duct of filters. A subset of a differential pretorsion theory is a subquantale of this quantale.

Saved in:
Bibliographic Details
Published in:Algebra and Discrete Mathematics
Date:2007
Main Author: Melnyk, I.
Format: Article
Language:English
Published: Інститут прикладної математики і механіки НАН України 2007
Online Access:https://nasplib.isofts.kiev.ua/handle/123456789/152386
Tags: Add Tag
No Tags, Be the first to tag this record!
Journal Title:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:On quantales of preradical Bland filters and differential preradical filters / I. Melnyk // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 108–122. — Бібліогр.: 16 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-152386
record_format dspace
spelling Melnyk, I.
2019-06-10T17:29:15Z
2019-06-10T17:29:15Z
2007
On quantales of preradical Bland filters and differential preradical filters / I. Melnyk // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 108–122. — Бібліогр.: 16 назв. — англ.
1726-3255
2000 Mathematics Subject Classification:20F05, 20E05, 57M07.
https://nasplib.isofts.kiev.ua/handle/123456789/152386
We prove that the set of all Bland preradical filters over an arbitrary differential ring form a quantale with respect to meets where the role of multiplication is played by the usual Gabriel pro-duct of filters. A subset of a differential pretorsion theory is a subquantale of this quantale.
en
Інститут прикладної математики і механіки НАН України
Algebra and Discrete Mathematics
On quantales of preradical Bland filters and differential preradical filters
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title On quantales of preradical Bland filters and differential preradical filters
spellingShingle On quantales of preradical Bland filters and differential preradical filters
Melnyk, I.
title_short On quantales of preradical Bland filters and differential preradical filters
title_full On quantales of preradical Bland filters and differential preradical filters
title_fullStr On quantales of preradical Bland filters and differential preradical filters
title_full_unstemmed On quantales of preradical Bland filters and differential preradical filters
title_sort on quantales of preradical bland filters and differential preradical filters
author Melnyk, I.
author_facet Melnyk, I.
publishDate 2007
language English
container_title Algebra and Discrete Mathematics
publisher Інститут прикладної математики і механіки НАН України
format Article
description We prove that the set of all Bland preradical filters over an arbitrary differential ring form a quantale with respect to meets where the role of multiplication is played by the usual Gabriel pro-duct of filters. A subset of a differential pretorsion theory is a subquantale of this quantale.
issn 1726-3255
url https://nasplib.isofts.kiev.ua/handle/123456789/152386
citation_txt On quantales of preradical Bland filters and differential preradical filters / I. Melnyk // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 108–122. — Бібліогр.: 16 назв. — англ.
work_keys_str_mv AT melnyki onquantalesofpreradicalblandfiltersanddifferentialpreradicalfilters
first_indexed 2025-11-25T22:45:08Z
last_indexed 2025-11-25T22:45:08Z
_version_ 1850570607282880512
fulltext Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 4. (2007). pp. 108 – 122 c© Journal “Algebra and Discrete Mathematics” On quantales of preradical Bland filters and differential preradical filters Ivanna Melnyk Communicated by V. V. Kirichenko Dedicated to Professor V. V. Kirichenko on the occasion of his 65th birthday Abstract. We prove that the set of all Bland preradical filters over an arbitrary differential ring form a quantale with respect to meets where the role of multiplication is played by the usual Gabriel pro-duct of filters. A subset of a differential pretorsion theory is a subquantale of this quantale. Introduction Rings considered in calculus are often equipped with additive maps that have properties of the operation of derivation. They include the ring of infinite differentiable real functions, the ring of all integer functions, the field of meromorphic functions and many others. That is why the study of rings equipped with derivations and, more generally, modules equipped with derivations is an important part of differential algebra. Recently Bland [2] has initiated the investigation of a new type of hered- itary tor-sion theories over such rings, which he called differential torsion theories. The investigation continued in [16], where such torsion theories were called Bland torsion theories. Another type of differential torsion theories was considered by O. L. Horbachuk, M. Ya. Komarnytskyi in [9]. The purpose of the present paper is to reveal the interrelation be- tween these types of torsion theories and to investigate the lattices of 2000 Mathematics Subject Classification: 20F05, 20E05, 57M07. Key words and phrases: differential ring, quantale, differential preradical filter, differential preradical Bland filter, differential torsion theory, Bland torsion theory. Jo u rn al A lg eb ra D is cr et e M at h .I. Melnyk 109 both Bland pretorsion theories and differential pretorsion theories from the point of view of quantale theory. In fact, we are dealing with more general functors than the functors of a torsion part, namely with the kernel functors, the general theory of which is developed in [6]. The concept of quantale [15] goes back to 1920’s, when W. Krull, followed by R. P. Dilworth and M.Ward, considered a lattice of ideals equipped with multiplication. The term ’quantale’ itself (short for ’quan- tum locale’) was suggested by C.J. Mulvey. A quantale is a complete lattice L satisfying the law a · ( ∨ i∈I bi) = ∨ i∈I(a · bi) for all a, bi ∈ L, where I is an index set. Note that, in this definition, we can substitute the meets by the joins, so that we obtain the dual quantale with respect to meets. It is well known that the set of all ideals of any ring is a complete lattice on which an operation of multiplication of ideals is defined. It satisfies an infinite distributive law, so the lattice of all ideals of the ring is a quantale. This quantale is intensively exploited in the paper [3]. By analogy, the set of all preradical filters of left ideals of an arbitrary ring is a complete lattice, moreover, an operation of Gabriel multiplication of preradical filters is defined. Hence, the lattice of preradical filters is, in fact, a quantale with respect to meets, see [5]. We wish to discover, which sets of differential pretorsion theories in the sense of Bland and pretorsion theories in the sense O.Horbachuk and M.Komarnytskyi over differential rings form a quantale. Preliminaries Throughout the paper, all rings are assumed to be associative with non- zero identity. All modules are unital left modules. The word ’ideal’, with- out any further comments, signifies a two-sided ideal. R − Mod denotes the category of left R-modules and module homomorphisms. Also, if I is a left ideal of the ring R and S ⊆ R, then the set (I : S) = {r ∈ R|rS ⊆ I} is a left ideal of R. In particular, when S = {a}, where a ∈ R, (I : a) denotes the left ideal of R given by {r ∈ R|ra ∈ I}. In the paper a standard ring-theoretic terminology will be used, following [8], [11]. Now we provide the basic definitions and facts on differential algebra, for more details, see [10]. An ordinary differential ring (or a δ-ring for short) is a pair (R, δ), where R is a ring and δ : R → R is a map, called a derivation on R, satisfying the following conditions: 1) δ(r + s) = δ(r) + δ(s), 2) δ(rs) = δ(r)s + rδ(s), Jo u rn al A lg eb ra D is cr et e M at h .110 Differential filters and Bland filters where in both equalities r and s are arbitrary elements of the ring R. If ∆ = {δi, i = 1, ..., n} is the set of pairwise commutative derivations on R, then the ring R together with the set ∆ is said to be a partial differential ring (a ∆-ring for short). A left partial differential R-module (a D-module for short) over a ∆- ring (R, ∆) is a pair (M, D), where M is an R -module and D is a set of pairwise commutative maps di : M → M , i = 1, 2, ..., n, called derivations on the module M consistent with δi, i = 1, ..., n (∆-derivations on M for short), each of which satisfies the following conditions for module derivations: 1) di(m + n) = di(m) + di(n) and 2) di(rm) = rdi(m) + δi(r)m for any m, n ∈ M , r ∈ R. Let (R, ∆) be a ∆-ring and let (M, D) be a left D-module over (R, ∆). A left ideal I of the ring R is called a ∆-ideal (short for differential ideal) if δ (I) ⊂ I for each δ ∈ ∆. Similarly, a submodule N of the module M is a D-submodule (short for differential submodule) if d (N) ⊂ N for each d ∈ D. A differential module M is called differentially simple if it has no differential submodules other than zero and itself (though it may have nontrivial non-differential submodules). Sometimes it may be useful to consider so called simple differential modules, i. e. simple modules, in the ordinary sense, equipped with some derivations. A differential submodule N of a differential module M is called a maximal differential submodule if there is no proper differential submodule between N and M . It is equivalent for the d-factor module M/N to be differentially simple, and the latter is equivalent for the module M/N to be simple when considered as a left module over the ring DR of linear differential operators of the differential ring R. The intersection of all maximal differential submodules of the left D-module M is a differential Jackobson radical of M , denoted by Jd(M). If M has no maximal differential submodules, then assume that Jd(M) = M. In case when M is a left regular differential module (i.e., M = RR ), we obtain the notion of a differential Jackobson radical of a differential ring R. It is clear that J(R) ⊆ Jd(R), where Jd(R) is the intersection of those left ideals which are maximal among the differential ideals of R. The differential Jacobson radical is vastly examined in many publications (see, e.g. [7]). A D-homomorphism of D-modules is a usual homomorphism of mod- ules which commutes with the operation of taking a derivative element with respect to each derivation, and an exact sequence of D-modules is D-exact if the linking maps are d-homomorphisms. The class of all left D-modules is, in fact, a category, where the role of morphisms is played Jo u rn al A lg eb ra D is cr et e M at h .I. Melnyk 111 by D-homomorphisms. This category is called the category of differ- ential modules and denoted by R − Dmod. The set of all differential homomorphisms from the D-module M to the D-module N is denoted by DhomR(M, N). Given a differential ring R together with the set of pairwise commu- tative derivations ∆ = {δ1, δ2, ..., δn} and a is an arbitrary element of R, we denote by a(∞) the set of all elements obtained from a by apply- ing a finite number of differential operators δ1, δ2, ..., δn, not necessarily distinct. Symbolically it is designated by a(∞) = = {( δ ik1 k1 δ ik2 k2 ...δ iks ks ) (a) |k1, ..., ks = 1, ..., n; ik1 , ..., iks ∈ N ∪ {0} } . Evidently, for any left differential ideal I and for any element a ∈ R, the ideal ( I : a(∞) ) is differential. It is easy to check that (( I : a(∞) ) : b(∞) ) = ( I : (ab)(∞) ) for every a, b ∈ R. Since the study of differential modules over the differential ring R is equivalent to the study of usual modules over the ring of linear differential operators DR with the coefficients of R, we must distinguish, which coef- ficient ring is considered in the context. The following functorial diagram will be helpful DR − Mod Θ // Ψ ''PPPPPPPPPPPP R − Mod, R − Dmod, Φ 77ooooooooooo where Θ is an inclusion functor, Ψ is the identity functor and Φ is a forgetful functor. 1. Differential preradical filters and preradical Bland filters Let (R, ∆) be a differential ring, ∆ = {δ1, δ2, ..., δn}, δiδj = δjδi, i, j = 1, ..., n. A nonempty collection F of left differential ideals of the differential ring R is said to be a differential preradical filter of R (see [9]) if the following conditions hold: DF1. If I ∈ F and I ⊆ J , where J is a left differential ideal of R, then J ∈ F ; Jo u rn al A lg eb ra D is cr et e M at h .112 Differential filters and Bland filters DF2. If I ∈ F and I ∈ F , then I ⋂ J ∈ F ; DF3. If I ∈ F , then (I : a(∞)) ∈ F for each a ∈ R. If a differential preradical filter F satisfies an extra condition DF4. If I ⊆ J with J ∈ F and (I : a(∞)) ∈ F for all a ∈ J , then I ∈ F , then the filter F is called a differential radical filter. Further we will call such filters HK-filters (differential radical filters in the sense of Horbachuk-Komarnytskyi ). When the derivatives δi , i = 1, ..., n are trivial, we obtain the defini- tion of the usual preradical and Gabriel filter. A preradical filter E of left ideals of the differential ring (R, ∆) is called a preradical Bland filter [2] if the following condition holds: For every I ∈ E , there exists a J ∈ E such that δi(J) ⊆ I for each i = 1, . . . , n. A Gabriel filter of left ideals of the differential ring R is called a radical Bland filter if it is the Bland filter as a preradical filter. The set of all Bland preradical filters will be denoted by FilB(R). Bland remarked that all Gabriel filters over a commutative ring are differential; in our terms, it means that all Gabriel filters over such a ring are radical Bland filters [2]. In that paper the examples confirming the importance of radical Bland filters are given. We now prove some propositions which make the Bland’s remark more accurate and show that there are plenty of Bland filters. In fact, it is much more difficult to give an example of the Gabriel filter of a differential ring which is not a radical Bland filter. Remind that a Gabriel filter which has a basis consisting of two-sided ideals is called symmetric (or bounded) (see [5]). Proposition 1. Any symmetric preradical filter (Gabriel filter) of an arbitrary differential ring is a preradical (radical) Bland filter. If a pre- radical filter (Gabriel filter) of left ideals of a differential ring R, not necessarily symmetric, has a basis consisting of left differential ideals, then it is a preradical (radical) Bland filter. Proof. The proof of the part of the proposition concerning preradical filters is not difficult. Thus, we prove it for Gabriel filters. Let E be a symmetric Gabriel filter and let I be a left ideal of the ring R which belong to the filter E . Then there exists a two-sided ideal K ∈ E such that K ⊆ I. Since E is multiplicatively closed, K2 ∈ E and δi(K 2) ⊆ I for every i = 1, ..., n. Hence, E is a Bland filter. Assume now that the Gabriel filter E has a basis B consisting of left differential ideals. We again consider an arbitrary left ideal I of E . Choose K ∈ B such that K ⊆ I. Since K is differential, then δi(K) ⊆ I, and we are done. Jo u rn al A lg eb ra D is cr et e M at h .I. Melnyk 113 The following Proposition shows that each HK-filter generates some Bland filter, necessarily unique, and that the inverse correspondence is not one-to-one. Proposition 2. Every differential HK-filter F of the ∆-ring R is a basis for some Gabriel filter EF of the ring R, which obviously is a radical Bland filter called a radical Bland filter generated by the HK-filter E. Moreover, the set BE of all differential left ideals which belong to some fixed radical Bland filter E is a basis for some HK-filter FE of the ∆-ring R. In this case different HK-filters correspond to different radical Bland filters. The proof of this statement involves a routine check of the correspond- ing conditions, therefore it is omitted. 2. Differential preradicals and kernel functors A preradical in the category R − Dmod is a functor σ : R − Dmod → R − Dmod such that: 1. σ(M) is differential submodule in M for each M ∈ R − Dmod; 2. For each differential homomorphism f : M → N we have f(σ(M)) ⊆ σ(N). Denote by R−Dpr the complete big lattice of all differential prerad- icals in R − Dmod. The most natural example of a preradical is the functor (differential Jacobson radical) JD : R − Dmod → R − Dmod, where JD : M 7→ JD(M). There are following classical operations in R − Dpr, namely ∧, ∨, · which are defined as follows, for σ, τ ∈ R − Dpr and M ∈ R − Dmod: (σ ∧ τ) (M) = σ (M) ∩ τ (M) , (σ ∨ τ) (M) = σ (M) + τ (M) , (σ · τ) (M) = σ (τ (M)) . The meet ∧ and the join ∨ can be defined for arbitrary families C of differential preradicals as follows: ∧{r ∈ C} (M) = ∩{r (M) |r ∈ C } , ∨{r ∈ C} (M) = ∑ {r (M) |r ∈ C }. Notice that for each M ∈ R − Dmod, {r (M) |r ∈ C } is a set. The operation · is called a product. It is well known that r1 · r2 6 r1 ∧ r2 6 r1 ∨ r2. Jo u rn al A lg eb ra D is cr et e M at h .114 Differential filters and Bland filters All these operations are associative and order-preserving. The functor σ ∈ R − Dpr is a left exact differential preradical if for each short exact sequence 0 → L f −→ M g −→ N → 0 the sequence 0 → r (L) σ(f) −→ r (M) σ(g) −→ r (N) is exact. The functor σ ∈ R−Dpr is a differential radical if σ (M/σ (M)) = 0 for each M ∈ R − Dmod. For any σ ∈ R − Dpr, we will use the following four classes of differ- ential modules: Tσ = {M ∈ R − Dmod |σ (M) = M } ; Fσ = {M ∈ R − Dmod |σ (M) = 0} ; T̄σ = {σ (M) |M ∈ R − Dmod} ; Fσ = {M/σ (M) |M ∈ R − Dmod} . Recall that σ is differentially idempotent if and only if Tσ = T̄σ, σ is differential radical if and only if Fσ = F̄σ. The functror σ is a left exact differential preradical if it is differentially idempotent and its differential pretorsion class Tσ is closed under taking differential submodules. The functor σ is a differential radical if and only if it is radical and its differ- ential pretorsion-free class Fσ is closed under taking differential quotient modules. Let σ, τ, η ∈ R −Dpr, {σα}α ⊆ R −Dpr, M ∈ R −Dmod. Then the following properties hold: 1. σ 6 τ ⇒ σ ∨ (τ ∧ η) = τ ∧ (σ ∨ η)(Modular law); 2. If {σα}α is a directed family, then τ ∧ (∨ασα) = ∨α (τ ∧ σα); 3. (∧ασα) τ = ∧α (σατ) ; 4. (∨ασα) τ = ∨α (σατ) . The classes of idempotent differential preradicals is closed under tak- ing arbitrary joins, and the classes of differential radicals and left exact differential preradicals are closed under taking arbitrary meets. A differential preradical σ : R − Dmod → R − Dmod is called a differential kernel functor if for every M ∈ R−Dmod and any differential submodule N of M , σ(N) = N ⋂ σ(M). For example, by putting in correspondence to each differential mod- ules M its differential socle SocD(M) = ∑ {P |P is a differentially simple module }, we obtain the kernel functor of taking the differential socle. Another example of a differential kernel functor is σF , obtained by using the differential preradical Bland filter or HK-filter F . Jo u rn al A lg eb ra D is cr et e M at h .I. Melnyk 115 Let σ be a differential kernel functor. Then the module M ∈ R − Dmod is said to be σ-torsion if σ(M) = M , and it is called σ-torsion- free if σ(M) = 0. A (pre)torsion theory is, in fact, defined by giving the class of torsion objects and the class of torsion-free objects. Denote by KD(R) the set of all differential kernel functors, and by ID(R) the set of all idempotent differential kernel functors. Then for an arbitrary ring R the following inclusion is satisfied: {0,∞} ⊆ ID(R) ⊆ KD(R). To an arbitrary differential kernel functor σ there is associated a pre- radical filter TD(σ) of the ring R consisting of left ideals I such that R/I is a submodule of some differential σ-torsion module. Now we state the lemma to be proved in the ordinary way. Lemma 1. The filter TD(σ) is preradical Bland filter. Proof. It is enough to prove the condition, defined by means of deriva- tions. Consider an arbitrary left ideal I ∈ TD(σ). Then there exists such a σ- torsion differential module M and an element m ∈ M that Annl(m) = I. Since M is σ-torsion, Annl(m ′) = J ∈ TD(σ). Take K = I ∩ J ∈ TD(σ). Then 0 = di(Km) ⊇ δi(K)m + Km′ = δi(K)m. It means that δi(K) ⊆ Annl(m) = I for each i = 1, ..., n. The kernel functor σ on category R−Mod will be called an extension of differential kernel functor σ if the functorial diagram R − Dmod σ // Φ �� R − Dmod Φ �� R − Mod σ // R − Mod is commutative. The set of all kernel functors σ which are extensions of some differential kernel functors will be denoted by Ke(R). The map Γ : σ 7→ TD(σ) is not one-to-one as the following simple example shows. Example. Let k be any universal differential field of characteristic 0. Then there only defined three different differential kernel functors on the category of all k-linear differential spaces k−Dmod, and all these kernel functors are, in fact, idempotent. However, there are only two kernel functors on the category of linear spaces k − Mod, therefore there exist only two trivial Bland filters over the ring k. In case when all the derivations δi, i = 1, ..., n, are zero, the notions introduced above transform into the well known notions of the torsion Jo u rn al A lg eb ra D is cr et e M at h .116 Differential filters and Bland filters theory over rings with no additional structures. In the corresponding notations the index D is omitted. For additional information on kernel functors see [6], [14]. The differential left module M will be called uniformly differential if each of its cyclic submodules is differential. A usual socle of any differen- tial module is uniformly differential. It is clear that the sum of any two uniformly differential submodules is a uniformly differential. Since the union of any chain of uniformly differential submodules is a uniformly differential module, then each differential module contain the largest uni- formly differential submodule U(M) (by Zorn’s lemma). It is easy to check that the functor U(M) is an idempotent differential kernel functor. A differential kernel functor will be called a kernel HK-functor if for each differential module M the inclusion σ(M) ⊆ U(M) holds. It is clear that for every HK-filter F the differential kernel functor σF is HK-functor. The following theorem shows that the converse is true. Theorem 1. There exists a one-to-one correspondence between (idempo- tent) kernel functors on the category R − Mod, which are the extensions of some differential kernel functor defined on R − Dmod, and (radical) preradical Bland filters of the ring R . Proof. Let σ ∈ K(R) and there is σ ∈ KD(R) for which σ |R−Dmod= σ. Then, by Lemma 1, the filter TD(σ) is a Bland filter. Thus, the map TD : Ke(R) → FilB(R) is well defined. Conversely, if F ∈ filB(R), then the kernel functor σF : R − Mod → R − Mod has the property: σF (M) ∈ R−Dmod for every M ∈ R−Dmod (this can by proved as in [2], p. 3). It means that σF |R−Dmod= σ. Therefore, the map ϑ : FilB(R) → Ke(R), were ϑ : F 7→ σF ∈ Ke(R) is well defined. Now we compute the compositions ϑ ◦ TD and TD ◦ ϑ. For the kernel functor σ : R−mod → R−mod, which is the extension of some differential kernel functor σ, we have (ϑ ◦ TD)(σ)=ϑ(TD(ϑ))=τ ∈ KD(R). If M is some left R-module, then τ(M) = {x ∈ M |∃I ∈ ID(σ), Ix = 0} = σ(M). Thus ϑ ◦ TD = 1|Ke(R). Conversely, if F ∈ FilB(R) then (TD ◦ ϑ)(F) = TD(θ(F)) = Jo u rn al A lg eb ra D is cr et e M at h .I. Melnyk 117 = {I|I ∈ L(R),∃M ∈ R − Dmod,∃x ∈ M, Annl(x) = I, M ∈ Tϑ(F)} = = {I|∃M ∈ R − Dmod,∃x ∈ M, Annl(x) ∈ F , Annl(x) = I} = F . Thus TD ◦ ϑ = 1|FilB(R). In the case of the idempotent kernel functor, the result is the conse- quence of the Lemma 1.5 in [2]. Theorem 2. There exists a one-to-one correspondence between (idempo- tent) kernel HK-functors on the category R − Dmod and (radical) HK- filters of the ring R. Proof. Let σ be an (idempotent) kernel HK-functor. Denote by Eσ the set of all left differential ideals of the ring R, for which there exists a uniformly differential σ-torsion module M and an element x ∈ M such that Ix = 0. Then all the conditions of the definition of (idempotent) HK-filters are satisfied (i.e., Eσ is a HK-filter). Thus, we have the map ϕ : KHK(R) → HKFil(R), where ϕ : σ 7→ Eσ. Now we show that ϕ is bijective. If σ 6= τ , σ, τ ∈ KHK(R) then there is a differential module M 6= 0 such that σ(M) 6= τ(M). Since σ(M) is uniformly differential, we can select an x ∈ M such that σ(Rx) 6= τ(Rx) and Annl(x) ∈ LD(R). Then Annl(x) ∈ Eτ�Eσ or Annl(x) ∈ Eσ�Eτ and we proved the injectivity of ϕ. The surjectivity of this map is evident, since for every HK-filter F ∈ HKFil(R) there is a kernel functor σF , defined by the rule σF (M) = {x ∈ M |Annl(x) ∈ F}, and it is a HK-functor over the ring R. Further- more, ϕ(σF ) = F . The set K(R) may be partially ordered by putting σ 6 τ if and only if σ(M) ⊆ τ(M) for all M ∈ R − Mod. Moreover, the set of differential kernel functors KD(R) is a complete lattice. In addition a product of kernel functors can be defined as follows: given σ and τ in KD(R), let A and B be their corresponding differential pretorsion classes. Consider the extension of B by A, i. e., the collection C = {M ∈ R−Dmod| there exists an exact sequence 0 −→ A −→ M −→ B −→ 0 with A ∈ A and B ∈ B}. Then C is a differential pretorsion class and we let τσ denote the corresponding differential kernel functor. In terms of preradical filters, this product amounts to defining F1F2 = {I ∈ R|∃J ∈ F2 such that I ⊆ J and (I : a) ∈ F1 for all a ∈ J}. It can be verifies that if I ∈ F1 and J ∈ F2, then IJ ∈ F1F2. With the natural order given by the inclusion of filters, the set of all preradical filters of left ideals Fil − R becomes structurally identical Jo u rn al A lg eb ra D is cr et e M at h .118 Differential filters and Bland filters to the set of all kernel functors K(R). The bottom of Fil(R) is the filter {R} which we will denote by 0; its corresponding kernel functor is the zero kernel functor, which we will denote by 0, i. e. the functor 0 : R − Mod → R − Mod defined by 0(M) = 0 for all M ∈ R − Mod. The largest element of Fil(R) is the set of all the left ideals of R; its corresponding kernel functor is the identity functor ∞, i. e. the functor ∞ : R−Mod → R−Mod defined by ∞(M) = M for all M ∈ R−Mod. Conversely, any collection C, closed under submodules, arbitrary di- rect sums, homomorphic images and extensions is of the form C = C(σ) for a unique σ ∈ I(R). Theorem 3. 1. The set of (idempotent) kernel functors σ ∈ K(R) on the category R−Mod which are the extensions of some (idempotent) dif- ferential kernel functors σ ∈ KD(R) defined on R−Dmod is an complete lattice. 2. The set of all (idempotent) kernel HK-functors is a complete lattice. Proof. 1. Let {σi}i∈I be a family of (idempotent) kernel functors on category R − Mod, where every σi is the extension of the (idempotent) differential kernel functor σi on R−Dmod. Then for every left R−module M we have ( ∧ i∈I σi)(M) = ⋂ i∈I σi(M). It is clear that ( ∧ i∈I σi) is an (idempotent) kernel functor. Denote by τ an (idempotent) differential kernel functor ∧ i∈I σi. If M is differential module then we have τ(M) = ( ∧ i∈I σi)(M) = ⋂ i∈I σi(M) = ⋂ i∈I σi(M) = ( ∧ i∈I σi)(M). It follow that ∧ i∈I σi is the extension of τ . By analogy, we prove that ∨ i∈I σi also exists and is the extension of some σ ∈ KD(R). 2. For any family of kernel HK-functors {σi}i∈I , the functors ∧ i∈I σi and ∨ i∈I σi are defined as above. We prove that they are HK-functors. If M ∈ R − Dmod then σi(M) is uniformly differential for all i ∈ I. It follows that ⋂ i∈I σi(M) and ⋃ i∈I σi(M) also are uniformly differentials. Thus, ∧ i∈I σi(M) and ∨ i∈I σi(M) are HK-functors. 3. Lattices of differential pretorsion theories and Bland pretorsion theories A differential torsion theory τ for the category R−Dmod is a pair (T, F ) of classes of differential R-modules such that 1. T ∩ F = 0; 2. If M → N → 0 is an differentially exact sequence in R − Dmod and M ∈ T, then N ∈ T; 3. If 0 → M → N is an differentially exact sequence in R − Dmod and N ∈ F, then M ∈ F; Jo u rn al A lg eb ra D is cr et e M at h .I. Melnyk 119 4. For each differential R-module M there exists a short differentially exact sequence 0 → T → M → F → 0 in R − Dmod for which T ∈ T and F ∈ F. Remind that a class T is a torsion class for a differential torsion theory τ if and only if it is closed under differential quotient modules, direct sums and extensions. A class F is a torsion-free class for τ if and only if F is closed under differential submodules, direct products and extensions. The differential modules in T are called τ -torsion, and the ones in F are τ -torsion-free. A differential torsion theory is called hereditary if T is closed under differential submodules, and cohereditary if F is closed under differential factor modules. A differential torsion theory is hereditary if and only if a torsion-free class F is closed under differential injective envelopes. Every differential R-module has a unique and the largest τ -torsion differential submodule given by tτ (M) = ∑ N ∈ S N , where S is the set of all τ -torsion differential submodules of the differential module M . In case when all the derivation are trivial we obtain the definition of a torsion theory for the category R − Mod. A torsion theory τ for R − Mod is called a Bland torsion theory if it is hereditary and the filter Eτ = {I|I is a left ideal of the ring R and R/I ∈ T} determined by τ is a Bland filter. In [2], Lemma 1.5) Bland has proved the following Proposition: Proposition 3. For a hereditary torsion theory τ on R − Mod the fol- lowing properties are equivalent: (1) Eτ is a Bland filter; (2) For each left R-module M and for each x ∈ τ(M) there is such I ∈ Eτ that δ(I) ⊆ (0 : x); (3) For each R-module M and for every derivation d defined on M , d(tτ (M)) ⊆ tτ (M). This proposition shows that a torsion theory in the category R−Mod is differential if and only if for each differential module M , d(tτ (M)) is its differential submodule (i.e., τ ∈ Ke(R)). Proposition 4. The intersection and union of an arbitrary family of preradical (radical) Bland filters of left ideals of the differential ring (R, ∆) is a preradical (radical) Bland filter of left ideals of (R, ∆). Proof. The assertions are direct consequences of the Theorems 1 and 2. Since the intersection of an arbitrary family of Bland filters is a Bland filter, we may see that the set of all Bland filters has the structure of a Jo u rn al A lg eb ra D is cr et e M at h .120 Differential filters and Bland filters complete lattice, where the meet and the join of Bland filters are defined in the usual way. We will also need the following fact. Proposition 5. If E1 and E2 are preradical (radical) Bland filters of left ideals of the differential ring (R, ∆), then their product E1 · E2 is a preradical (radical) Bland filter of left ideals of (R, ∆). Proof. It is well known from [5] that the product of preradical filters is a preradical filter. We only need to show that the filter E1 · E2 satisfies the second condition. Let I ∈ E1 · E2. Then there exists such J ∈ E2 that I ⊆ J and (I : a) ∈ E1 for all a ∈ J . The there exist such left ideals K ∈ E2 and Ka ∈ E1 with a ∈ J that δi(K) ⊆ J and δi(Ka) ⊆ (I : a) for each i = 1, 2, . . . , n. Consider the left ideal T = I ⋂ K of R. It is clear that δi(T ) ⊆ I. Since T ⊆ J ⋂ K and J ⋂ K ∈ E2, the inclusion (T : a) = (I ⋂ K : a) ⊇ (I : a), for every a ∈ J ⋂ K, follows that (T : a) ∈ E2. It means that T ∈ E1 · E2. 4. Quantales of Bland and HK-filters A quantale Q is a complete lattice with an associative binary multiplica- tion ∗ satisfying x ∗ ( ∨ i∈I xi ) = ∨ i∈I (x ∗ xi) and ( ∨ i∈I xi ) ∗ x = ∨ i∈I (xi ∗ x) for all x, xi ∈ Q, i ∈ I, I is a set. 1 denotes the greatest element of the quantale Q, 0 is the smallest element of Q. A quantale Q is said to be unital if there is an element u ∈ Q such that u ∗ a = a ∗ u = a for all a ∈ Q. A meet of the preradical filters F1 and F2 is the preradical filter F1 ∧ F2 which is the intersection of F1 and F2. A join of preradical filters F1 and F2 is the least preradical filter F1 ∨ F2 which contain both F1 and F2. A product of preradical filters F1 and F2 is a set F1 · F2 of those left differential ideals of the differential ring R for which there exists an ideal H ∈ F2 such that I ⊆ H and (I : a(∞)) ∈ F1 for all a ∈ J . By a subquantale of a quantale Q is meant a subset K closed under joins and multiplication. Jo u rn al A lg eb ra D is cr et e M at h .I. Melnyk 121 Proposition 6. The set of all HK-filters of the differential ring R forms a quantale with respect to meets. Proof. If I ∈ F · ( ∧ i∈Ω Fi) then there exists such J ∈ ∧ i∈Ω Fi that I ⊆ J and (I : a(∞)) ∈ F for all a ∈ J . Then for all i ∈ Ω J ∈ Fi and I ⊆ J and (I : a(∞)) ∈ F , so for all i ∈ Ω I ∈ F · Fi, that is I ∈ ∧ i∈Ω(F · Fi). If I ∈ ( ∧ i∈Ω Fi) · F then there exists such JF that I ⊆ J and (I : a(∞)) ∈ ∧ i∈Ω Fi for all a ∈ J . Then for all i ∈ Ω J ∈ F and I ⊆ J and (I : a(∞)) ∈ Fi, so for all i ∈ Ω I ∈ Fi · F , that is I ∈ ∧ i∈Ω(Fi · F). Check that F ∗ (∧i∈IFi) = ∧i∈I (F ∗ Fi) and (∧i∈IFi) ∗ F = = ∧i∈I (Fi ∗ F). We have that F ∗ (∧i∈IFi) = {I|∃H ∈ ∧i∈IFiI ⊆ H∀a ∈ H(I : a) ∈ F} = {I|∃H ∈ ⋂ i∈I FiI ⊆ H∀a ∈ H(I : a) ∈ F} = {I|∃∀i ∈ IH ∈ FiI ⊆ H∀a ∈ H(I : a) ∈ F} = ∧i∈I (F ∗ Fi) . Theorem 4. The set of all preradical Bland filters of left ideals of the differential ring (R, ∆) is a quantale with respect to meets, which is a subquantale of the quantale of all preradical filters of the differential ring (R, ∆). Proof. As proved by J.Golan, for all preradical filters Ei ∈ Fill(R), i ∈ I, E ∈ Fill(R) the following equality is valid: E · ( ∧ i∈I Ei) = ∧ i∈I (E · Ei) (see [5], Proposition 3.13, p.37). It means that the complete lattice Fill(R) is a quantale with respect to meets. (In fact, J.Golan stated his results in terms of semirings). It follows that, in order to finish the proof, it is enough to apply the Proposition 5 and the Theorems 1, 3. Acknowledgements. The author is grateful to professor M. Ya. Ko- marnytskyi for supervising the work and helpful remarks and comments. References [1] O. D. Artemovych, Differentially simple rings: survey, Matematychni Studii, vol. 23, No. 2, 2005, 115-128. [2] P. E. Bland, Differential torsion theory, Journal of Pure and pplied Algebra, 204, 2006, pp. 1-8. [3] F.Borceux, R.Cruciani, Generic Representation Theorem for Non-commutative Rings, J. Algebra , 167, 1994, pp. 291-308. [4] J. S. Golan, Torsion Theories, in: Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 29, Longman Scientific & Technical, Harlow, 1986, John Wiley & Sons Inc., New York. [5] J. S. Golan, Linear topologies on a ring: an overview, in: Pitman Research Notes in Mathematics Series 159, Longman Scientific & Technical, Harlow, 1987, John Wiley & Sons Inc., New York. Jo u rn al A lg eb ra D is cr et e M at h .122 Differential filters and Bland filters [6] O. Goldman, Rings and modules of quotients, J. Algebra, 13, 1969, pp. 10-47. [7] H. E. Gorman, Differential rings and modules, Scripta mathematica, Vol. 29, No. 1-2, 1979. [8] M. Hazewinkel, N. Gubareni, V. V. Kirichenko Algebras, Rings and Modules Volume 1, Kluwer academic publisher, New York, Boston, Dortrecht, London, Moscow. 2004. [9] O. L. Horbachuk, M. Ya. Komarnytskyi, On differential torsions, In: Teoretical and applied problems of algebra and differential equations, Kyev, Naukova Dumka, 1977 (in Russian). [10] I. Kaplansky, Introduction to differential algebra, in: Graduate Texts in Mathe- matics, vol. 189, Springer-Verlag, New York, 1999. [11] T. Y. Lam, Lectures on Modules and Rings, in: Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. [12] A. V. Mikhaliev, E. V. Pankratiev, Differential and difference algebra, Itogi nauki i tehniki, Seriya Algebra. Topology. Geometry, Vol. 25, 1987, pp. 67- . [13] A. P. Mishina, L.A. Skorniakov, Abelian groups and modules, Nauka, Moskov, 1969. [14] F. van Oystaeyen, Primer spectra of non-commutative algebra, Lect. Notes in Math. 444, Springer-Verlag, Berlin, 1975. [15] J. Paseka, J. Rosicky, Quantales, in: B. Coecke, D. Moore, A. Wilce, (Eds.), Current Research in Operational Quantum Logic: Algebras, Categories and Lan- guages, Fund. Theories Phys., vol. 111, Kluwer Academic Publishers, 2000, pp. 245-262. [16] L. Vas, Differerntiability of torsion theories, Journal of Pure and Applied Algebra, 210, 2007, pp. 847-853. Contact information I. Melnyk Algebra and Logic Department, Mechanics and Mathematics Faculty, Ivan Franko Na- tional University of Lviv, 1 Universytetska Str., Lviv, 79000, Ukraine E-Mail: ivannamelnyk@yahoo.com Received by the editors: 15.05.2007 and in final form 15.05.2007.

Similar Items