Generalizations of semicoprime preradicals

Generalizations of semicoprime preradicals

This article introduces the notions quasi-co-n-absorbing preradicals and semi-co-n-absorbing preradicals, generalizing the concept of semicoprime preradicals. We study the concepts quasi-co-n-absorbing submodules and semi-co-n-absorbing submodules and their relations with quasi-co-n-absorbing prer...

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Hauptverfasser: Darani, A.Y., Mostafanasab, H.
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spelling nasplib_isofts_kiev_ua-123456789-1552402025-02-09T14:05:39Z Generalizations of semicoprime preradicals Darani, A.Y. Mostafanasab, H. This article introduces the notions quasi-co-n-absorbing preradicals and semi-co-n-absorbing preradicals, generalizing the concept of semicoprime preradicals. We study the concepts quasi-co-n-absorbing submodules and semi-co-n-absorbing submodules and their relations with quasi-co-n-absorbing preradicals and semi-co-n-absorbing preradicals using the lattice structure of preradicals. The authors would like to thank the referee for the careful reading of the manuscript and all the suggestions that improved the paper. 2016 Article Generalizations of semicoprime preradicals / A.Y. Darani, H. Mostafanasab // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 214-238. — Бібліогр.: 25 назв. — англ. 1726-3255 2010 MSC:16N99, 16S99, 06C05, 16N20. https://nasplib.isofts.kiev.ua/handle/123456789/155240 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description This article introduces the notions quasi-co-n-absorbing preradicals and semi-co-n-absorbing preradicals, generalizing the concept of semicoprime preradicals. We study the concepts quasi-co-n-absorbing submodules and semi-co-n-absorbing submodules and their relations with quasi-co-n-absorbing preradicals and semi-co-n-absorbing preradicals using the lattice structure of preradicals.
format Article
author Darani, A.Y.
Mostafanasab, H.
spellingShingle Darani, A.Y.
Mostafanasab, H.
Generalizations of semicoprime preradicals
Algebra and Discrete Mathematics
author_facet Darani, A.Y.
Mostafanasab, H.
author_sort Darani, A.Y.
title Generalizations of semicoprime preradicals
title_short Generalizations of semicoprime preradicals
title_full Generalizations of semicoprime preradicals
title_fullStr Generalizations of semicoprime preradicals
title_full_unstemmed Generalizations of semicoprime preradicals
title_sort generalizations of semicoprime preradicals
publisher Інститут прикладної математики і механіки НАН України
publishDate 2016
url https://nasplib.isofts.kiev.ua/handle/123456789/155240
citation_txt Generalizations of semicoprime preradicals / A.Y. Darani, H. Mostafanasab // Algebra and Discrete Mathematics. — 2016. — Vol. 21, № 2. — С. 214-238. — Бібліогр.: 25 назв. — англ.
series Algebra and Discrete Mathematics
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fulltext Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 21 (2016). Number 2, pp. 214–238 © Journal “Algebra and Discrete Mathematics” Generalizations of semicoprime preradicals Ahmad Yousefian Darani and Hojjat Mostafanasab Communicated by R. Wisbauer Abstract. This article introduces the notions quasi-co- n-absorbing preradicals and semi-co-n-absorbing preradicals, gen- eralizing the concept of semicoprime preradicals. We study the concepts quasi-co-n-absorbing submodules and semi-co-n-absorbing submodules and their relations with quasi-co-n-absorbing preradi- cals and semi-co-n-absorbing preradicals using the lattice structure of preradicals. 1. Introduction The notion of 2-absorbing ideals of commutative rings was introduced by Badawi in [2], where a proper ideal I of a commutative ring R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. Anderson and Badawi [1] generalized the concept of 2-absorbing ideals to n-absorbing ideals. According to their definition, a proper ideal I of R is called an n-absorbing (resp. strongly n-absorbing) ideal if whenever x1 · · · xn+1 ∈ I for x1, . . . , xn+1 ∈ R (resp. I1 · · · In+1 ⊆ I for ideals I1, . . . , In+1 of R), then there are n of the xi’s (resp. n of the Ii’s) whose product is in I. In [24], the concept of 2-absorbing ideals was generalized to submodules of a module over a commutative ring. A proper submodule N of an R-module M is said to be a 2-absorbing submodule of M if whenever a, b ∈ R and m ∈ M with abm ∈ N , then ab ∈ (N :R M) or am ∈ N or bm ∈ N . For more studies concerning 2010 MSC: 16N99, 16S99, 06C05, 16N20. Key words and phrases: lattice, preradical, quasi-co-n-absorbing, semi-co-n- absorbing. A. Yousefian Darani, H. Mostafanasab 215 2-absorbing (submodules) ideals we refer to [3],[9],[24],[25]. In [13], Raggi et al. introduced the concepts of prime preradicals and prime submodules over noncommutative rings, and Raggi, Ríos and Wisbauer [18], studied the dual notions of these, coprime preradicals and coprime submodules. A generalization of prime preradicals and submodules, “2-absorbing preradi- cals and submodules” was investigated by Yousefian and Mostafanasab in [23]. In [14], Raggi et al. defined and investigated semiprime preradicals, and Mostafanasab and Yousefian [10], studied the concepts of quasi-n- absorbing and semi-n-absorbing preradicals. Raggi et al. [11] defined the notions of semicoprime preradicals and submodules. In this paper, we introduce the concepts of “quasi-co-n-absorbing preradicals” and “semi- co-n-absorbing preradicals”. As well we investigate“quasi-co-n-absorbing submodules” and “semi-co-n-absorbing submodules” in this paper. 2. Preliminaries Throughout this paper R is an associative ring with nonzero identity, and R-Mod denotes the category of all the unitary left R-modules. We denote by R-simp a complete set of representatives of isomorphism classes of simple left R-modules. For M ∈ R-Mod, we denote by E(M) the injective hull of M . Let U, N ∈ R-Mod, we say that N is generated by U (or N is U -generated) if there exists an epimorphism U (Λ) → N for some index set Λ. Dually, we say that N is cogenerated by U (or N is U -cogenerated) if there exists a monomorphism N → UΛ for some index set Λ. Also, we say that an R-module X is subgenerated by M (or X is M -subgenerated) if X is a submodule of an M -generated module. The category of M -subgenerated modules (the Wisbauer category) is denoted σ[M ] (see [21]). A preradical over the ring R is a subfunctor of the identity functor on R-Mod. Denote by R-pr the class of all preradicals over R. There is a natural partial ordering in R-pr given by σ � τ if σ(M) 6 τ(M) for every M ∈ R-Mod. It is proved in [15] that with this partial ordering, R-pr is an atomic and co-atomic big lattice. The smallest and the largest elements of R-pr are denoted, respectively, 0 and 1. Let M ∈ R-Mod. Recall ([5] or [15]) that a submodule N of M is called fully invariant if f(N) 6 N for each R-homomorphism f : M → M . In this paper, the notation N 6fi M means that “N is a fully invariant submodule of M”. Obviously the submodule K of M is fully invariant if and only if there exists a preradical τ of R-Mod such that K = τ(M). If N 6 M , then the preradicals αM N and ωM N are defined as follows: For K ∈ R-Mod, 216 Generalizations of semicoprime preradicals 1) αM N (K) = ∑ {f(N)|f ∈ HomR(M, K)}. 2) ωM N (K) = ⋂ {f−1(N)|f ∈ HomR(K, M)}. Notice that for σ ∈ R-pr and M, N ∈ R-Mod we have that σ(M) = N if and only if N 6fi M and αM N � σ � ωM N . We have also that if K 6 N 6 M with K, N 6fi M , then αM K � αM N and ωM K � ωM N . The atoms and coatoms of R-pr are, respectively, {α E(S) S | S ∈ R-simp} and {ωR I | I is a maximal ideal of R} (See [15, Theorem 7]). There are four classical operations in R-pr, namely, ∧, ∨, · and : which are defined as follows. For σ, τ ∈ R-pr and M ∈ R-Mod: 1) (σ ∧ τ)(M) = σM ∩ τM , 2) (σ ∨ τ)(M) = σM + τM , 3) (στ)(M) = σ(τM) and 4) (σ : τ)(M) is determined by (σ : τ)(M)/σM = τ(M/σM). The meet ∧ and join ∨ can be defined for arbitrary families of preradicals as in [15]. The operation defined in (3) is called product, and the operation defined in (4) is called coproduct. It is easy to show that for σ, τ ∈ R-pr, στ � σ ∧ τ � σ ∨ τ � (σ : τ). It is clear that in R-pr the operations (1)-(3) are associative, and in [22] it was shown that the coproduct “ :′′ is associative. Notice the fact that coproduct of preradicals preserves order on both sides, see [8, Remark 2.1]. We denote σσ · · · σ (n times) by σn and (σ : σ : · · · : σ) (n times) by σ[n]. Recall that σ ∈ R-pr is an idempotent if σ2 = σ, while σ is a radical if σ[2] = σ. Note that σ is a radical if and only if, σ(M/σ(M)) = 0 for each M ∈ R-Mod. We say that σ is nilpotent if σn = 0 for some n > 1, while σ is unipotent if σ[n] = 1 for some n > 1. Using the preradical ωM N , in the papers [6], [7] and [18], the following operation was introduced and studied: ω-coproduct of submodules K, N 6 M : (K :M N) = (ωM K : ωM N )(M). Henceforward, for brevity, (K : N) is written instead of (K :M N). For any σ ∈ R-pr, we will use the following class of R-modules: Tσ = {M ∈ R − Mod | σ(M) = M}. Let σ ∈ R-pr. By [18, Theorem 8.2], the following classes of modules are closed under taking arbitrary meets and arbitrary joins: Ae = {τ ∈ R-pr | τσ = σ} and At = {τ ∈ R-pr | (σ : τ) = 1}. As in [16], we define, for σ ∈ R-pr, the following preradicals: • e(σ) = ∧ {τ ∈ Ae} the equalizer of σ; A. Yousefian Darani, H. Mostafanasab 217 • t(σ) = ∧ {τ ∈ At} the totalizer of σ. Clearly e(σ)σ = σ and (σ : t(σ)) = 1. For undefined notions we refer the reader to [13,15–17]. In [18], Raggi et al. defined the notions of coprime preradicals and coprime submodules as follows: Let σ ∈ R-pr. σ is called coprime in R-pr if σ 6= 0 and for any τ, η ∈ R-pr, σ � (τ : η) implies that σ � τ or σ � η. Let M ∈ R-Mod and let N 6 M be a nonzero fully invariant submodule of M . The submodule N is said to be coprime in M if whenever K, L are fully invariant submodules of M with N 6 (K : L), then N 6 K or N 6 L. Also, Raggi et al. [11] defined a preradical σ semicoprime in R-pr if σ 6= 0 and for any τ ∈ R-pr, σ � (τ : τ) implies that σ � τ . They said that a nonzero fully invariant submodule N of M is semicoprime in M if whenever K is a fully invariant submodule of M with N 6 (K : K), then N 6 K. In special case, M is called a coprime (resp. semicoprime) module if M is a coprime (resp. semicoprime) submodule of itself. Yousefian and Mostafanasab in [22] defined the notions of co-2-absorb- ing preradicals and co-2-absorbing submodules. The preradical σ ∈ R-pr is called co-2-absorbing if σ 6= 0 and, for each η, µ, ν ∈ R-pr, σ � (η : µ : ν) implies that σ � (η : µ) or σ � (η : ν) or σ � (µ : ν). More generally, a preradical 0 6= σ in R-pr is said to be a co-n-absorbing preradical if whenever σ � (η1 : η2 : · · · : ηn+1) for η1, η2, . . . , ηn+1 ∈ R-pr, there are i1, i2, . . . , in ∈ {1, 2, . . . , n + 1} such that i1 < i2 < · · · < in and σ � (ηi1 : ηi2 : · · · : ηin). They denoted by R-co-ass the class of all R-modules M that the operation ω-coproduct is associative over fully invariant submodules of M , i.e., for any fully invariant submodules K, N, L of M , ((K : N) : L) = (K : (N : L)). Let M ∈ R-co-ass and K be a fully invariant submodule of M . Then (K : K : · · · : K) (n times) is simply denoted by K[n]. By Proposition 5.4 of [7], we can see that if an R-module M is injective and artinian, then M ∈ R-co-ass. Let M ∈ R-co-ass and N a nonzero fully invariant submodule of M . The submodule N is said to be co-2-absorbing in M if whenever J, K, L are fully invariant submodules of M with N 6 (J : K : L), then N 6 (J : K) or N 6 (J : L) or N 6 (K : L). The generalization of co-2-absorbing submodules is that, the submodule N is said co-n-absorbing in M if whenever N 6 (K1 : K2 : · · · : Kn+1) for fully invariant submodules K1, K2, . . . , Kn+1 of M , there are i1, i2, . . . , in ∈ {1, 2, . . . , n + 1} such that i1 < i2 < · · · < in and N 6 (Ki1 : Ki2 : · · · : Kin). An R-module M is called a co-n-absorbing module if M is a co-n-absorbing submodule of itself. 218 Generalizations of semicoprime preradicals We say that a preradical 0 6= σ ∈ R-pr is called a quasi-co-n-absorbing preradical if whenever σ � (µ[n] : ν) for µ, ν ∈ R-pr, then σ � µ[n] or σ � (µ[n−1] : ν). A preradical 0 6= σ ∈ R-pr is called a semi-co-n-absorbing preradical if whenever σ � µ[n+1] for µ ∈ R-pr, then σ � µ[n]. Let M ∈ R- co-ass. We say that a nonzero fully invariant submodule N of M is quasi- co-n-absorbing in M if for every fully invariant submodules K, L of M , N 6 (K[n] : L) implies that N 6 K[n] or N 6 (K[n−1] : L). A nonzero fully invariant submodule N of M is called semi-co-n-absorbing in M if for every fully invariant submodule K of M , N 6 K[n+1] implies that N 6 K[n]. An R-module M satisfies the ω-property if (τ(M) :M η(M)) = (τ : η)(M) for every τ, η ∈ R-pr, see [22]. We recall the definition of relative epi-projectivity (see [20]). Let M and N be modules. N is said to be epi-M -projective if, for any submodule K of M , any epimorphism f : N → M K can be lifted to a homomorphism g : N → M Proposition 1 ([22, Proposition 2.9 (1)]). Let M ∈ R-Mod. If for any fully invariant submodule K of M , M K is epi-M-projective, then M has the ω-property. In the next sections we frequently use the following proposition. Proposition 2 ([12, Proposition 1.2]). Let {Mγ}γ∈I and {Nγ}γ∈I be families of modules in R-Mod such that for each γ ∈ I, Nγ 6 Mγ. Let N = ⊕ γ∈I Nγ, M = ⊕ γ∈I Mγ, N ′ = ∏ γ∈I Nγ and M ′ = ∏ γ∈I Mγ. (1) If N 6fi M , then for each γ ∈ I, Nγ 6fi Mγ and αM N = ∨ γ∈I α Mγ Nγ . (2) If N ′ 6fi M ′, then for each γ ∈ I, Nγ 6fi Mγ and ωM ′ N ′ = ∧ γ∈I ω Mγ Nγ . 3. Quasi-co-n-absorbing preradicals Suppose that m, n are positive integers with n > m. A preradical σ 6= 0 is called a quasi-co-(n, m)-absorbing preradical if whenever σ � (µ[n−1] : ν) for µ, ν ∈ R-pr, then σ � µ[m] or σ � (µ[m−1] : ν). Proposition 3. Let σ ∈ R-pr and let m > 0. The following conditions are equivalent: (1) σ is quasi-co-(n, m)-absorbing for every n > m; (2) σ is quasi-co-(n, m)-absorbing for some n > m; (3) σ is quasi-co-m-absorbing. Proof. (1)⇒(2) Is trivial. A. Yousefian Darani, H. Mostafanasab 219 (2)⇒(3) Assume that σ is quasi-co-(n, m)-absorbing for some n > m. Let σ � (µ[m] : ν) for some µ, ν ∈ R-pr. Since m 6 n−1, then (µ[m] : ν) � (µ[n−1] : ν) and so σ � (µ[n−1] : ν). Therefore σ � µ[m] or σ � (µ[m−1] : ν). Consequently σ is quasi-co-m-absorbing. (3)⇒(1) Suppose that σ is quasi-co-m-absorbing and get n > m. Let σ � (µ[n−1] : ν) for some µ, ν ∈ R-pr. Therefore σ � (µ[m] : (µ[n−1−m] : ν)). Hence σ � µ[m] or σ � (µ[m−1] : (µ[n−1−m] : ν)) = (µ[n−2] : ν). Repeating this method implies that σ � µ[m] or σ � (µ[m−1] : ν). Thus σ is quasi-co-(n, m)-absorbing. Remark 1. Let σ ∈ R-pr. (1) σ is coprime if and only if σ is quasi-co-1-absorbing if and only if σ is co-1-absorbing. (2) If σ is quasi-co-n-absorbing, then it is quasi-co-i-absorbing for all i > n. (3) If σ is coprime, then it is quasi-co-n-absorbing for all n > 1. (4) If σ is quasi-co-n-absorbing for some n > 1, then there exists the least n0 > 1 such that σ is quasi-co-n0-absorbing. In this case, σ is quasi-co-n-absorbing for all n > n0 and it is not quasi-co-i-absorbing for n0 > i > 0. Proposition 4. Let C be a family of coprime preradicals. Then ∨ σ∈C σ is a quasi-co-i-absorbing preradical for every i > 2. Proof. Let τ = ∨ σ∈C σ. By Remark 1(2), it is sufficient to show that τ is a quasi-co-2-absorbing preradical. Suppose that τ � (µ[2] : ν) for some µ, ν ∈ R-pr. Since every σ ∈ C is coprime and σ � (µ[2] : ν), then σ � µ or σ � ν. Hence τ � (µ : ν), and so we conclude that τ is a quasi-co-2-absorbing preradical. Let ζ = ∨ {αS S | S ∈ R-simp}. Note that for every R-module M , ζ(M) = Soc(M). As in [14], ζ is called the socle preradical. Also, let κ = {α R/I R/I | I a maximal ideal of R}. We call κ the ultrasocle preradical, see [11]. As a direct consequence of Proposition 4 we have the following result. Proposition 5. ζ, κ are quasi-co–i-absorbing preradicals for every i > 2. Proof. By [18, p. 57], for each simple R-module S, αS S is coprime. Also, for every maximal ideal I of R, α R/I R/I is a coprime preradical, [11, Remark 6]. Then by Proposition 4, the claim holds. 220 Generalizations of semicoprime preradicals Proposition 6. If R is a semisimple Artinian ring, then every nonzero preradical σ ∈ R-pr is a quasi-co-i-absorbing preradical for every i > 2. Proof. Suppose that R is a semisimple Artinian ring. According to [18, Proposition 3.2], every atom α E(S) S is a coprime preradical. On the other hand [15, Theorem 11] implies that σ = ∨ {α E(S) S | S ∈R-simp, α E(S) S � σ}. Therefore every nonzero preradical σ in R-pr is quasi-co-i-absorbing for every i > 2, by Proposition 4. Remark 2. Let S1, S2, . . . , Sn+1 ∈ R-simp be distinct. Then by Proposi- tion 4, αS1 S1 ∨αS2 S2 ∨· · ·∨α Sn+1 Sn+1 is a quasi-co-i-absorbing preradical in R-pr for every i > 2. But, [22, Proposition 3.6] implies that αS1 S1 ∨αS2 S2 ∨· · ·∨α Sn+1 Sn+1 is not a co-n-absorbing preradical. This remark shows that the two concepts of quasi-co-n-absorbing preradicals and of co-n-absorbing preradicals are different in general. Corollary 1. If R is a ring such that every quasi-co-n-absorbing prerad- ical in R-pr is co-n-absorbing, then |R-simp| 6 n. Notice the fact that coproduct of preradicals preserves order on both sides. Proposition 7. Let R be a ring. The following statements are equivalent: (1) for every µ, ν ∈ R-pr, (µ[n] : ν) = µ[n] or (µ[n] : ν) = (µ[n−1] : ν); (2) for every σ1, σ2, . . . , σn+1 ∈ R-pr, (σ1 : σ2 : · · · : σn+1) � (σ1 ∨ σ2 ∨ · · · ∨ σn)[n] or (σ1 : σ2 : · · · : σn+1) � ((σ1 ∨ σ2 ∨ · · · ∨ σn)[n−1] : σn+1); (3) every preredical 0 6= σ ∈ R-pr is quasi-co-n-absorbing. Proof. (1)⇒(2) If σ1, σ2, . . . , σn+1 ∈ R-pr, then by part (1) we have that, (σ1 : σ2 : · · · : σn+1) � ((σ1 ∨ σ2 ∨ · · · ∨ σn)[n] : σn+1) = (σ1 ∨ σ2 ∨ · · · ∨ σn)[n], or (σ1 : σ2 : · · · : σn+1) � ((σ1 ∨ σ2 ∨ · · · ∨ σn)[n] : σn+1) = ((σ1 ∨ σ2 ∨ · · · ∨ σn)[n−1] : σn+1). A. Yousefian Darani, H. Mostafanasab 221 (2)⇒(1) For preradicals µ, ν ∈ R-pr, we have from (2), (µ[n] : ν) � ( n times︷ ︸︸ ︷ µ ∨ · · · ∨ µ)[n] = µ[n] or (µ[n] : ν) � (( n times︷ ︸︸ ︷ µ ∨ · · · ∨ µ)[n−1] : ν) = (µ[n−1] : ν). Thus we have that (µ[n] : ν) = µ[n] or (µ[n] : ν) = (µ[n−1] : ν). (1)⇔(3) Is evident. In the next proposition we use (µ1 : · · · : µ̂i : · · · : µn+1) when the i-th term is excluded from (µ1 : · · · : µn+1). Proposition 8. Let 0 6= σ ∈ R-pr be an idempotent radical. (1) If σ is such that for any µ, ν ∈ R-pr, we have µ ∨ ν � σ � (µ[n] : ν) ⇒ [σ � µ[n] or σ � (µ[n−1] : ν)], then σ is quasi-co-n-absorbing. (2) If σ is such that for any µ1, µ2, . . . , µn+1 ∈ R-pr, we have µ1 ∨ µ2 ∨ · · · ∨ µn+1 � σ � (µ1 : µ2 : · · · : µn+1) ⇒ [σ � (µ1 : · · · : µ̂i : · · · : µn+1), for some 1 6 i 6 n + 1], then σ is a co-n-absorbing preradical. Proof. (1) Let σ 6= 0 be an idempotent radical that satisfies the hypothesis in part (1). Let σ � (τ[n] : λ) for some τ, λ ∈ R-pr. Then, by [15, Theorem 8(3)] we have τσ ∨ λσ � σ = σ2 � (τ[n] : λ)σ = (τ[n]σ : λσ) = ((τσ)[n] : λσ). So, by hypothesis we have σ � (τσ)[n] = τ[n]σ � τ[n] or σ � ((τσ)[n−1] : λσ) = (τ[n−1] : λ)σ � (τ[n−1] : λ). Therefore σ is quasi-co-n-absorbing. (2) The proof is similar to that of (1). Proposition 9. Let C be a chain of quasi-co-n-absorbing preradicals, that is, a subclass of quasi-co-n-absorbing preradicals which is linearly ordered. Then ∨ σ∈C σ is a quasi-co-n-absorbing preradical. 222 Generalizations of semicoprime preradicals Proof. Let τ = ∨ σ∈C σ and assume that τ � (µ[n] : ν) for some µ, ν ∈ R- pr. If σ � µ[n] for each σ ∈ C, then τ � µ[n]. If there exists σ0 ∈ C such that σ0 � µ[n], then σ � µ[n] for each σ0 � σ. Since all preradicals in C are quasi-co-n-absorbing, it follows that σ � (µ[n−1] : ν) for each σ0 � σ. Thus σ � (µ[n−1] : ν) for each σ ∈ C, so that τ � (µ[n−1] : ν). Consequently, we deduce that τ is a quasi-co-n-absorbing preradical. Proposition 10. If σi is a quasi-co-ni-absorbing preradical in R-pr for every 1 6 i 6 k, then σ1 ∨σ2 ∨· · ·∨σk is a quasi-co-n-absorbing preradical for n = n1 + · · · + nk. Proof. For k = 1 there is nothing to prove. Then, suppose that k > 1. Assume that σ1 ∨ σ2 ∨ · · · ∨ σk � (µ[n] : ν) for some µ, ν ∈ R-pr. Notice that for every 1 6 i 6 k, σi � (µ[n] : ν) = (µ[ni] : µ[n−ni] : ν). Then, for every 1 6 i 6 k, either σi � µ[ni] or σi � (µ[ni−1] : µ[n−ni] : ν) = (µ[n−1] : ν), because σi is quasi-co-ni-absorbing. On the other hand, for every 1 6 i 6 k, µ[ni] � µ[n−1] and so µ[ni] � (µ[n−1] : ν). Hence σ1 ∨ σ2 ∨ · · · ∨ σk � (µ[n−1] : ν) which shows that σ1 ∨ σ2 ∨ · · · ∨ σk is a quasi-co-n-absorbing preradical. Proposition 11. Let σ1, σ2, . . . , σt ∈ R-pr. (1) If σ1 is a quasi-co-n-absorbing preradical and σ2 is a quasi-co-m- absorbing preradical for m 6 n, then σ1 ∨ σ2 is a quasi-co-(n + 1)- absorbing preradical. (2) If σ1, σ2, . . . , σt are quasi-co-n-absorbing preradicals, then σ1 ∨ σ2 ∨ · · · ∨ σt is a quasi-co-(n + t − 1)-absorbing preradical. (3) If σi is a quasi-co-ni-absorbing preradical for every 1 6 i 6 t with n1 < n2 < · · · < nt and t > 2, then σ1 ∨ σ2 ∨ · · · ∨ σt is a quasi-co- (nt + 1)-absorbing preradical. Proof. (1) Let µ, ν ∈ R-pr be such that σ1 ∨ σ2 � (µ[n+1] : ν). Since σ1 is quasi-co-n-absorbing and σ1 � (µ[n] : µ : ν), then either σ1 � µ[n] or σ1 � (µ[n−1] : µ : ν) = (µ[n] : ν). Also, σ2 is quasi-co-m-absorbing and σ2 � (µ[m] : µ[n+1−m] : ν), so either σ2 � µ[m] or σ2 � (µ[m−1] : µ[n+1−m] : ν) = (µ[n] : ν). There are four cases. Case 1. Assume that σ1 � µ[n] and σ2 � µ[m]. Then σ1 ∨ σ2 � µ[n]. Case 2. Assume that σ1 �µ[n] and σ2 �(µ[n] : ν). Then σ1 ∨ σ2 � (µ[n] : ν). Case 3. Assume that σ1 �(µ[n] : ν) and σ2 �µ[m]. Then σ1 ∨ σ2 � (µ[n] : ν). Case 4. Assume that σ1 � (µ[n] : ν) and σ2 � (µ[n] : ν). Then σ1 ∨ σ2 � (µ[n] : ν). Hence σ1 ∨ σ2 is quasi-co-(n + 1)-absorbing. A. Yousefian Darani, H. Mostafanasab 223 (2) We use induction on t. For t = 1 there is nothing to prove. Let t > 1 and assume that for t−1 the claim holds. Then σ1 ∨σ2 ∨· · ·∨σt−1 is quasi-co-(n + t − 2)-absorbing. Since σt is quasi-co-n-absorbing, then it is quasi-co-(n+ t−2)-absorbing, by Remark 1(2). Therefore σ1 ∨σ2 ∨· · ·∨σt is quasi-co-(n + t − 1)-absorbing, by part (1). (3) Induction on t: For t = 3 apply parts (1) and (2). Let t > 3 and suppose that for t − 1 the claim holds. Hence σ1 ∨ σ2 ∨ · · · ∨ σt−1 is quasi-co-(nt−1 + 1)-absorbing. We consider the following cases: Case 1. Let nt−1 +1 < nt. In this case σ1 ∨σ2 ∨· · ·∨σt is quasi-co-(nt +1)- absorbing, by part (1). Case 2. Let nt−1 + 1 = nt. Thus σ1 ∨ σ2 ∨ · · · ∨ σt is quasi-co-(nt + 1)- absorbing, by part (2). Case 3. Let nt−1 + 1 > nt. Then σ1 ∨ σ2 ∨ · · · ∨ σt is quasi-co-(nt−1 + 2)- absorbing, by part (1). Since nt−1 + 2 6 nt + 1, then σ1 ∨ σ2 ∨ · · · ∨ σt is quasi-co-(nt + 1)-absorbing. Proposition 12. Let σ ∈ R-pr be a radical. If σ is quasi-co-n-absorbing, then e(σ) is quasi-co-n-absorbing. Proof. Assume that σ is quasi-co-n-absorbing, and let e(σ) � (µ[n] : ν) for some µ, ν ∈ R-pr. Then σ = e(σ)σ � (µ[n] : ν)σ � ((µσ)[n] : νσ). Since σ is quasi-co-n-absorbing and radical, [15, Theorem 8(3)] implies that either σ � (µσ)[n] = µ[n]σ � µ[n] or σ � ((µσ)[n−1] : νσ) = (µ[n−1] : ν)σ � (µ[n−1] : ν). Consequently e(σ) is quasi-co-n-absorbing. Definition 1. For τ, ρ ∈ R-pr define the totalizer of ρ relative to τ as tτ (ρ) = ∧ {η ∈ R-pr| (ρ : η) � τ}. Note that t1(ρ) = t(ρ). Proposition 13. Let τ ∈ R-pr. If τ is quasi-co-2-absorbing, then for each λ ∈ R-pr, either τ � λ[n] or tτ (λ[n]) = tτ (λ[n−1]). In particular, if 1 is a quasi-co-2-absorbing preradical, then for each λ ∈ R-pr, either λ[n] = 1 or t(λ[n]) = t(λ[n−1]). Proof. Suppose that τ is quasi-co-2-absorbing and let λ ∈ R-pr such that τ � λ[n]. If ν ∈ R-pr is such that τ � (λ[n] : ν), then τ � (λ[n−1] : ν), since σ is quasi-co-2-absorbing. Therefore tτ (λ[n−1]) � tτ (λ[n]). On the other hand λ[n−1] � λ[n] and so tτ (λ[n]) � tτ (λ[n−1]). Consequently tτ (λ[n]) = tτ (λ[n−1]). 224 Generalizations of semicoprime preradicals 4. Semi-co-n-absorbing preradicals Suppose that m, n are positive integers with n > m. A more general concept than semi-co-n-absorbing preradicals is the concept of semi-co- (n, m)-absorbing preradicals. A preradical σ 6= 0 is called a semi-co-(n, m)- absorbing preradical if whenever σ � µ[n] for µ ∈ R-pr, then σ � µ[m]. Note that a semicoprime preradical is just a semi-co-1-absorbing preradical. Theorem 1. Let σ ∈ R-pr and m, n be positive integers with n > m. (1) If σ is quasi-co-m-absorbing, then it is semi-co-(k, m)-absorbing for every k > m. (2) If σ is semi-co-(n, m)-absorbing, then it is semi-co-(i, m)-absorbing for every m < i < n, in particular it is semi-co-m-absorbing. (3) σ is semi-co-(n, m)-absorbing if and only if σ is semi-co-(n, k)- absorbing for each n > k > m if and only if σ is semi-co-(i, j)- absorbing for each n > i > j > m. (4) If σ is semi-co-(n, m)-absorbing, then it is semi-co-(nk, mk)-absorb- ing for every positive integer k. (5) If σ is semi-co-(n, m)-absorbing and semi-co-(r, s)-absorbing for some positive integers r > s, then it is semi-co-(nr, ms)-absorbing. Proof. (1) Is trivial. (2) Is easy. (3) Straightforward. (4) Suppose that σ is semi-co-(n, m)-absorbing. Let µ ∈ R-pr and let k be a positive integer such that σ � µ[nk]. Then σ � ( µ[k] ) [n] . Since σ is semi-co-(n, m)-absorbing, σ � ( µ[k] ) [m] = µ[mk], and so σ is semi-co- (nk, mk)-absorbing. (5) Assume that σ is semi-co-(n, m)-absorbing and semi-co-(r, s)- absorbing for some positive integers r > s. Let σ � µ[nr]. Since σ is semi-co-(n, m)-absorbing, then σ � µ[mr]; and since σ is semi-co-(r, s)- absorbing, σ � µ[ms]. Hence σ is semi-co-(nr, ms)-absorbing. Corollary 2. Let σ ∈ R-pr and n be a positive integer. (1) If σ is quasi-co-n-absorbing, then it is semi-co-n-absorbing. (2) Let t 6 n be an integer. If σ is semi-co-(n + 1, t)-absorbing, then it is semi-co-(nk + i, tk)-absorbing for all k > i > 1. (3) If σ is semi-co-n-absorbing, then it is semi-co-(nk + i, nk)-absorbing for all k > i > 1. A. Yousefian Darani, H. Mostafanasab 225 (4) If σ is semi-co-n-absorbing, then it is semi-co-(nk + j)-absorbing for all k > j > 0. (5) If σ is semi-co-n-absorbing, then it is semi-co-(nk)-absorbing for every positive integer k. (6) If σ is semicoprime, then it is semi-co-k-absorbing for every positive integer k. (7) If σ is semicoprime, then for every k > 1 and every µ ∈ R-pr, σ � µ[k] implies that σ � µ. (8) If σ is semi-co-n-absorbing, then it is semi-co-((n + 1)t, nt)-absorb -ing for all t > 1. (9) If σ is semicoprime, then it is quasi-co-k-absorbing for every k > 1. Proof. (1) By parts (1), (2) of Theorem 1. (2) Let σ be semi-co-(n + 1, t)-absorbing. Then by Theorem 1(4), σ is semi-co-(nk + k, tk)-absorbing, for every positive integer k. Hence by Theorem 1(2), σ is semi-co-(nk + i, tk)-absorbing for every k > i > 1. (3) In part (2) get t = n. (4) By part (3). (5) Is a special case of (4). (6) Is a direct consequence of (5). (7) By part (6). (8) By Theorem 1(5). (9) Assume that σ is semicoprime. Let σ � (µ[k] : ν) for some µ, ν ∈ R-pr and some k > 1. Then σ � (µ[k] : ν) � (µ : ν)[k]. Therefore σ � (µ : ν), by part (7). So σ is quasi-co-k-absorbing. In the following remark we prove Proposition 4 in another way. Remark 3. Clearly, an arbitrary join of a family of semicoprime (coprime) preradicals is semicoprime, and so it is quasi-co-k-absorbing for every k > 1, by Corollary 2(9). Proposition 14. Let σ1, σ2, . . . , σn ∈ R-pr. If for every 1 6 i 6 n, σi is a semicoprime preradical, then (σ1 : σ2 : · · · : σn) is a semi-co-n-absorbing preradical. In particular, if σ is a semicoprime preradical, then σ[n] is a semi-co-n-absorbing preradical. Proof. Apply Corollary 2(7). Lemma 1. Let σ ∈ R-pr. If σ[n+1] is a semi-co-n-absorbing preradical, then σ[n+1] = σ[n]. In particular, if σ[2] is a semicoprime preradical, then σ is radical. 226 Generalizations of semicoprime preradicals Proposition 15. Let σ ∈ R-pr, σ 6= 0 be an idempotent radical. If σ is such that for any µ ∈ R-pr, we have µ � σ � µ[n+1] ⇒ σ � µ[n], then σ is semi-co-n-absorbing. Proof. The proof is similar to that of Proposition 8(1). Proposition 16. Let σ1, σ2, . . . , σn ∈ R-pr be semi-co-2-absorbing pre- radicals. Then (σ1 : σ2 : · · · : σn) is a semi-co-(3n−1)-absorbing preradical. Proof. Suppose that (σ1 : σ2 : · · · : σn) � µ[3n] for some µ ∈ R-pr. For every 1 6 i 6 n, σi � µ[3n] = ( µ[3n−1] ) [3] and σi is semi-co-2-absorbing, then σi � ( µ[3n−1] ) [2] = µ[2·3n−1] = ( µ[2·3n−2] ) [3] . Again, since σi is semi- co-2-absorbing, we conclude that σi � µ[22·3n−2]. Repeating this method implies that σi � µ[2n]. So (σ1 : σ2 : · · · : σn) � µ[n2n]. On the other hand n2n 6 3n − 1. So (σ1 : σ2 : · · · : σn) � µ[3n−1] which shows that (σ1 : σ2 : · · · : σn) is semi-co-(3n − 1)-absorbing. Proposition 17. If σi is a semi-co-ni-absorbing preradical in R-pr for every 1 6 i 6 k, then σ1 ∨ σ2 ∨ · · · ∨ σk is a semi-co-(n − 1)-absorbing preradical for n = k∏ i=1 (ni + 1). Proof. Let µ ∈ R-pr be such that σ1 ∨ σ2 ∨ · · · ∨ σk � µ[n]. Thus for every 1 6 i 6 k, σi � ( µ[m] ) [ni+1] , where m = k∏ j=1, j 6=i (nj + 1). Since σi’s are semi-co-ni-absorbing, then, for each 1 6 i 6 k, σi � µ[nim]. Note that for every 1 6 i 6 k, nim 6 k∏ i=1 (ni + 1) − 1 = n − 1. So we have σi � µ[n−1] for every 1 6 i 6 k. Hence σ1∨σ2∨· · ·∨σk � µ[n−1] which implies that σ1 ∨ σ2 ∨ · · · ∨ σk is a semi-co-(n − 1)-absorbing preradical. Proposition 18. Let σ1, σ2 ∈ R-pr and m, n be positive integers. (1) If σ1 is quasi-co-m-absorbing and σ2 is semi-co-n-absorbing, then (σ1 : σ2) is semi-co-(n(m + 1) + m)-absorbing. (2) If σ1 is quasi-co-(2m)-absorbing and σ2 is semi-co-m-absorbing, then (σ1 : σ2) is semi-co-(m2 + 2m)-absorbing. Proof. (1) Suppose that (σ1 : σ2) � µ[(n+1)(m+1)] for some µ ∈ R-pr. Since σ1 is quasi-co-m-absorbing and σ1 � µ[(n+1)(m+1)], then σ1 � µ[m]. A. Yousefian Darani, H. Mostafanasab 227 On the other hand σ2 is semi-co-n-absorbing and σ2 � µ[(n+1)(m+1)], then σ2 � µ[n(m+1)]. Consequently (σ1 : σ2) � µ[n(m+1)+m], and so (σ1 : σ2) is semi-co-(n(m + 1) + m)-absorbing. (2) Suppose that (σ1 : σ2) � µ[(m+1)2] for some µ ∈ R-pr. Since σ1 is quasi-co-(2m)-absorbing and σ1 � µ[(m+1)2], then σ1 � µ[2m]. Since σ2 is semi-co-m-absorbing and σ2 � µ[(m+1)2], then σ2 � µ[m2]. Hence (σ1 : σ2) � µ[m2+2m] which shows that (σ1 : σ2) is semi-co-(m2 + 2m)- absorbing. Proposition 19. Let R be a ring. The following statements are equiva- lent: (1) for every preradical σ ∈ R-pr, σ[n+1] = σ[n]; (2) for all preradicals σ1σ2, . . . , σn+1 ∈ R-pr we have (σ1 : σ2 : · · · : σn+1) � (σ1 ∨ σ2 ∨ · · · ∨ σn+1)[n]; (3) every preredical 0 6= σ ∈ R-pr is semi-co-n-absorbing. Proof. (1)⇒(2) If σ1, σ2, . . . , σn+1 ∈ R-pr, then we get from (1), (σ1 : σ2 : · · · : σn+1) � (σ1 ∨σ2 ∨· · ·∨σn+1)[n+1] = (σ1 ∨σ2 ∨· · ·∨σn+1)[n]. (2)⇒(1) For a preradical σ ∈ R-pr, we have from (2), σ[n+1] � ( n+1 times︷ ︸︸ ︷ σ ∨ · · · ∨ σ)[n] = σ[n]. So we have that σ[n+1] = σ[n]. (1)⇔(3) Is clear. Remark 4. Let {σα}α∈I ⊆ R-pr. If σα is semi-co-n-absorbing for every α ∈ I, then ∨ α∈I σα is semi-co-n-absorbing. Proposition 20. Let σ ∈ R-pr be radical. If σ is semi-co-n-absorbing, then e(σ) is semi-co-n-absorbing. Proof. Is similar to the proof of Proposition 12. In Proposition 23 of [11], it was shown that σ0 := ∨ {σ ∈ R-pr | σ is semicoprime} is the unique greatest semicoprime preradical. Proposition 21. There exists in R-pr a unique greatest semi-co-n- absorbing preradical. 228 Generalizations of semicoprime preradicals Proof. Set σ0 (n) = ∨ {σ ∈ R-pr | σ is semi-co-n-absorbing}. By Remark 4, σ0 (n) is the greatest semi-co-n-absorbing preradical. By notation in the the proof of the previous proposition we have that σ0 (1) = σ0. Remark 5. As ζ � κ � σ0 are semicoprime preradicals, then ζ[n], κ[n], σ0 [n] are semi-co-n-absorbing preradicals, by Proposition 14. Therefore ζ[n] � κ[n] � σ0 [n] � σ0 (n). Proposition 22. The following statements hold: (1) σ0 = ∧ n>1 σ0 (n). (2) σ0 (n) � σ0 [nk] for every positive integer k. (3) σ[n] � σ0 (n) for every semicoprime preradical σ. Proof. (1) By Corollary 2(6) every semicoprime preradical is semi-co-n- absorbing for every n > 1. Then σ0 � σ0 (n) for every n > 1. (2) By Corollary 2(5). (3) By Proposition 14. In Proposition 26 of [11] it was shown that σ0 � ν0, where ν0 =∧ {τ | τ ∈ R-pr, τ is unipotent}. The following proposition is straightforward. Proposition 23. Suppose that ν (n) 0 := ∧ {τ[n] | τ ∈ R-pr, τ[n+1] = 1}. Then: (1) σ0 (n) � ν (n) 0 ; (2) ν0 � ν (1) 0 . Corollary 3. The following statements hold: (1) If ζ[n+1] = 1, then ζ[n] = κ[n] = σ0 [n] = σ0 (n) = ν (n) 0 ; (2) If ζ[2] = 1, then ζ = κ = σ0 = ν0 = ν (1) 0 . Proof. (1) By Remark 5 and Proposition 23 we have that ζ[n] � κ[n] � σ0 [n] � σ0 (n) � ν (n) 0 . If ζ[n+1] = 1, then ν (n) 0 � ζ[n], and so ζ[n] = κ[n] = σ0 [n] = σ0 (n) = ν (n) 0 . (2) By part (1) and [11, Corollary 27]. Proposition 24. For a ring R the following statements are equivalent: (1) For every µ ∈ R-pr, µ[n+1] = 1 implies that µ[n] = 1; A. Yousefian Darani, H. Mostafanasab 229 (2) 1 is a semi-co-n-absorbing preradical; (3) σ0 (n) = 1; (4) ν (n) 0 = 1. Proof. Is easy. For τ ∈ R-pr define C(n)(τ) = ∨ {σ ∈ R-pr | σ � τ, σ semi-co-n-absorbing}, which is the unique greatest semi-co-n-absorbing preradical less than or equal to τ . Notice that in [11], C(1) is denoted by C. Proposition 25. Let R be a ring. (1) σ0 (n) = C(n)(1) = ∨ τ∈R-pr C(n)(τ). (2) For each τ ∈ R-pr, C(n)(τ) � τ . (3) For each τ, σ ∈ R-pr we have τ � σ ⇒ C(n)(τ) � C(n)(σ). (4) For each τ ∈ R-pr, C(n)(τ[n+1]) = C(n)(τ[n]). (5) For each τ ∈ R-pr, τ is semi-co-n-absorbing if and only if τ = C(n)(τ). (6) {τ ∈ R-pr | τ is semi-co-n-absorbing} = Im C(n) = {C(n)(σ) | σ ∈ R-pr}. (7) [ C(n) ]2 = C(n). Thus, C(n) is a closure operator on R-pr. (8) For each family {τα}α∈I ⊆ R-pr, we have C(n)( ∧ α∈I τα) = C(n)( ∧ α∈I C(n)(τα)). (9) C(n) = ∧ k>1 C(nk), in particular C = ∧ k>1 C(k). (10) C(n)(σ[n+1]) = C(n)(σ[n]) = σ[n] for any semicoprime preradical σ. Proof. The proofs of (1), (2), (3), (5) and (6) is easy. (4) For any τ ∈ R-pr, part (3) implies that C(n)(τ[n]) � C(n)(τ[n+1]). Since C(n)(τ[n+1]) is semi-co-n-absorbing (by Remark 4) and C(n)(τ[n+1]) � τ[n+1], then C(n)(τ[n+1]) � τ[n]. Hence C(n)(τ[n+1]) � C(n)(τ[n]). So the equality holds. (7) Is a direct consequence of part (5). (8) The proof is similar to that of [11, Proposition 31](5). (9) By Corollary 2(5). (10) Apply Proposition 14 and parts (4), (5). 230 Generalizations of semicoprime preradicals Now consider the operator (−) in R-pr that assigns to each preradical σ the least radical over σ (see [19, p. 137]). Lemma 2. Let σ, τ ∈ R-pr be such that σ is radical and τ is semi-co-n- absorbing. Then: (1) C(n)(σ) � C(n)(σ) � σ. (2) C(n)(σ) = C(n)(C(n)(σ)). (3) τ � C(n)(τ) � τ . (4) τ = C(n)(τ). Proof. Similar to the proof of [11, Lemma 32]. Proposition 26. Let R be a ring. (1) The operator C(n)(−) defines an interior operator on the ordered class of radicals. (2) The operator C(n)((−)) defines a closure operator on the ordered class of semi-co-n-absorbing preradicals. Notice that the “open” radicals associated with the interior operator C(n)(−) are O (n) rad = {σ radical | σ = τ for some semi-co-n-absorbing τ}. The “closed” semi-co-n-absorbing preradicals associated with the closure operator C(n)((−)) are C(n) sca = {τ semi-co-n-absorbing | τ = C(n)(σ) for some radical σ}. The following result is immediate. Corollary 4. For a ring R the operators C(n)(−) and (−) restrict to mutually inverse maps between O (n) rad and C (n) sca. Definition 2. Let τ ∈ R-pr. Define C (n) 1 (τ) = ∧ {σ[n] | σ ∈ R-pr, τ � σ[n+1]}. Proposition 27. For a ring R the following conditions hold: (1) For each τ ∈ R-pr, C (n) 1 (τ) � τ[n]. (2) For each τ ∈ R-pr, τ is semi-co-n-absorbing if and only if τ � C (n) 1 (τ). (3) 1 is a semi-co-n-absorbing preradical if and only if C (n) 1 (1) = 1. A. Yousefian Darani, H. Mostafanasab 231 (4) Let τ, σ ∈ R-pr. If τ � σ, then C (n) 1 (τ) � C (n) 1 (σ). (5) For each family {τα}α∈I ⊆ R-pr, C (n) 1 ( ∧ α∈I τα) � ∧ α∈I C (n) 1 (τα) and ∨ α∈I C (n) 1 (τα) � C (n) 1 ( ∨ α∈I τα). Proof. The assertions have straightforward verifications. We apply an “Amitsur construction” to C (n) 1 as follows: Definition 3. Let τ ∈ R-pr. We define recursively the preradical C (n) λ (τ) for each ordinal λ as follows: (1) C (n) 0 (τ) = τ . (2) C (n) λ+1(τ) = C (n) 1 (C (n) λ (τ)). (3) If λ is a limit ordinal, then C (n) λ (τ) = ∧ β<λ C (n) β (τ). (4) C (n) Ω (τ) = ∧ λ ordinal C (n) λ (τ). Proposition 28. Let τ ∈ R-pr. Then the following statements are equiv- alent: (1) τ is semi-co-n-absorbing; (2) For each ordinal λ, τ � C (n) λ (τ); (3) C (n) Ω (τ) = τ . Proof. By Proposition 27 and using transfinite induction we have the claim. As is the case with C (n) 1 , all of the operators C (n) λ preserve order between preradicals. Proposition 29. Let τ, σ ∈ R-pr be such that τ � σ. Then: (1) For each ordinal λ, C (n) λ (τ) � C (n) λ (σ). (2) C (n) Ω (τ) � C (n) Ω (σ). Proposition 30. For each τ ∈ R-pr, C(n)(τ) � C (n) Ω (τ). Proof. Let τ ∈ R-pr. We use transfinite induction. First, note that C(n)(τ) � τ = C (n) 0 (τ). Assume that λ is an ordinal such that C(n)(τ) � C (n) λ (τ). Since C(n)(τ) is semi-co-n-absorbing, C(n)(τ) � C (n) 1 (C(n)(τ)) � C (n) 1 (C (n) λ (τ)) = C (n) λ+1(τ), by parts (2) and (4) of Proposition 27. If λ is a limit ordinal and C(n)(τ) � C (n) β (τ) for each β < λ, then C(n)(τ) � ∧ β<λ C (n) β (τ) = C (n) λ (τ). 232 Generalizations of semicoprime preradicals In the following result we give equivalent conditions for the equality C (n) Ω (τ) = C(n)(τ). Proposition 31. For each τ ∈ R-pr the following statements are equiv- alent: (1) C (n) Ω (τ) is semi-co-n-absorbing; (2) C (n) Ω (τ) � C (n) 1 (C (n) Ω (τ)); (3) For each ordinal λ we have C (n) Ω (τ) � C (n) λ (C (n) Ω (τ)); (4) C (n) Ω (C (n) Ω (τ)) = C (n) Ω (τ); (5) C (n) Ω (τ) = C(n)(τ). Proof. (1)⇒(2) By Proposition 27(2). (2)⇒(3) It follows by using transfinite induction on λ. (3)⇒(4) Is easy. (4)⇒(1) By Proposition 28. (1)⇒(5) Assume that C (n) Ω (τ) is semi-co-n-absorbing. Since C (n) Ω (τ)�τ , the definition of C(n)(τ) implies that C (n) Ω (τ) � C(n)(τ). On the other hand C(n)(τ) � C (n) Ω (τ), by Proposition 30. So the equality holds. (5)⇒(1) Is straightforward. 5. Quasi-co-n-absorbing and semi-co-n-absorbing submodules Remark 6. Let M ∈ R-co-ass and N be a nonzero fully invariant submodule of M . Then we have: (1) N is co-n-absorbing in M ⇒ N is quasi-co-n-absorbing in M ⇒ N is semi-co-n-absorbing in M . (2) N is a quasi-co-1-absorbing submodule of M if and only if N is a coprime submodule of M . (3) N is a semi-co-1-absorbing submodule of M if and only if N is a semicoprime submodule of M . Proposition 32. Let σ ∈ R-pr. If for every M ∈ R-Mod, σ(M) is a semicoprime submodule of M , then σ is a semicoprime preradical. Proof. By hypothesis, [11, Proposition 19] implies that αM σ(M) is a semi- coprime preradical. So σ = ∨ {αM σ(M) | M ∈ R-Mod} (see [17, Remark 1]) is a semicoprime preradical. Corollary 5. Let R be a ring. If every nonzero R-module is semicoprime, then 1 is a semicoprime preradical in R-pr. A. Yousefian Darani, H. Mostafanasab 233 Lemma 3 ([7, Lemma 2.5]). Let M ∈ R-Mod. Then for any submodules N, K of M , αM N+K = αM N ∨ αM K . Proposition 33. Let M ∈ R-Mod. Suppose that {Ni}i∈I is a family of semicoprime submodules of M . Then N = ∑ i∈I Ni is a semicoprime submodule of M . Proof. Let {Ni}i∈I be a family of semicoprime submodules of M . Then, by [11, Proposition 19], αM Nj ’s are semicoprime preradicals. Thus αM N = ∨ i∈I αM Ni is a semicoprime preradical. Again by [11, Proposition 19], N = ∑ i∈I Ni is a semicoprime submodule of M . Proposition 34. Let R be a ring and {Mi}i∈I be a family of semicoprime R-modules. Then M = ⊕ i∈I Mi is a semicoprime R-module. Proof. Since for every i ∈ I, Mi is a semicoprime R-module, then for every i ∈ I, αMi Mi is a semicoprime preradical, by [11, Proposition 19]. Therefore ∨ i∈I αMi Mi = αM M is a semicoprime preradical, and so again by [11, Proposition 19], M = ⊕ i∈I Mi is a semicoprime R-module. Proposition 35. For a ring R the following statements are equivalent: (1) R is a finite product of simple rings; (2) κ = 1; (3) 1 is a semicoprime preradical; (4) RR is a semicoprime R-module; (5) There exists a semicoprime R-module that is a generator in R-Mod. Proof. (1)⇔(2) By [11, Theorem 10]. (1)⇔(3) By [11, Theorem 29]. (3)⇔(4) Notice the fact that an R-module G is a generator in R-Mod if and only if αG G = 1. Since R is a generator in R-Mod, then αR R = 1. Now, use [11, Proposition 19]. (4)⇒(5) Is trivial. (5)⇒(3) See the proof of (3)⇔(4). Theorem 2. Let M ∈ R-co-ass and N a fully invariant submodule of M . Consider the following statements. (a) N is co-n-absorbing in M . (b) αM N is a co-n-absorbing preradical. Then (b) ⇒ (a), and if M satisfies the ω-property, then (a) ⇒ (b). 234 Generalizations of semicoprime preradicals Proof. The proof is similar to that of [22, Theorem 4.2]. Theorem 3. Let M ∈ R-co-ass and N a fully invariant submodule of M . Consider the following statements: (1) N is quasi-co-n-absorbing (resp. semi-co-n-absorbing) in M . (2) αM N is a quasi-co-n-absorbing (resp. semi-co-n-absorbing) preradical. Then (2) ⇒ (1), and if M satisfies the ω-property, then (1) ⇒ (2). Proof. (1) ⇒ (2) Assume that N is quasi-co-n-absorbing in M and that (η(M) : µ(M)) = (η : µ)(M) for every η, µ ∈ R-pr. Since N 6= 0 we have αM N 6= 0. Now let η, µ ∈ R-pr be such that αM N � (η[n] : µ). In this case we have N = αM N (M) 6 (η[n] : µ)(M) = (η(M)[n] : µ(M)). Since N is quasi-co-n-absorbing in M , by hypothesis we have that N 6 η(M)[n] = η[n](M) or N 6 (η(M)[n−1] : µ(M)) = (η[n−1] : µ)(M). It follows from [15, Proposition 5] that αM N � αM η[n](M) � η[n] or αM N � αM (η[n−1]:µ)(M) � (η[n−1] : µ), and so αM N is quasi-co-n-absorbing. (2) ⇒ (1) Assume that αM N is a quasi-co-n-absorbing preradical. Since αM N 6= 0, we have N 6= 0. Suppose that J, K are fully invariant submodules of M such that N 6 (J[n] : K). Then we have N 6 ( (ωM J )[n] : ωM K ) (M). By [15, Proposition 5], we get αM N � αM ((ωM J )[n]:ω M K )(M) � ( (ωM J )[n] : ωM K ) . Since αM N is quasi-co-n-absorbing, we have αM N � (ωM J )[n] or αM N �( (ωM J )[n] : ωM K ) . Therefore N = αM N (M) � (ωM J )[n](M) = J[n] or N = αM N (M) � ( (ωM J )[n] : ωM K ) (M) = (J[n−1] : K). Hence N is a quasi-co- n-absorbing submodule. A similar proof can be stated for semi-co-n- absorbing preradicals. Remark 7. Note that in Theorem 3, for the case n = 2 we can omit the condition M ∈ R-co-ass, by the definition of quasi-co-2-absorbing (semi-co-2-absorbing) submodules. Definition 4. Let M ∈ R-co-ass. We say that M is a quasi-co-n-absorbing (resp. semi-co-n-absorbing) module if M is a quasi-co-n-absorbing (resp. semi-co-n-absorbing) submodule of itself. A. Yousefian Darani, H. Mostafanasab 235 Corollary 6. Let M1, M2, . . . , Mt be injective Artinian R-modules. Sup- pose that Mi’s are quasi-co-n-absorbing modules that satisfy the ω-property. Then M = ⊕t i=1 Mi is a quasi-co-(n + t − 1)-absorbing R-module. Proof. Let M1, M2, . . . , Mt be quasi-co-n-absorbing R-modules. Then, by Theorem 3, αM1 M1 , αM2 M2 , . . . , αMt Mt are quasi-co-n-absorbing preradicals, and so αM M = αM1 M1 ∨ αM2 M2 ∨ · · · ∨ αMt Mt is a quasi-co-(n + t − 1)-absorbing preradical, by Proposition 11(2). Again by Theorem 3, M = ⊕t i=1 Mi is a quasi-co-(n + t − 1)-absorbing R-module. Corollary 7. Let R be a ring. The following statements hold: (1) If the preradical 1 is quasi-co-2-absorbing (resp.semi-co-2-absorbing), then every generator R-module is a quasi-co-2-absorbing (resp. semi- co-2-absorbing) R-module. (2) If R is a semisimple Artinian ring, then every R-module is quasi- co-i-absorbing for every i > 2. Proof. (1) Suppose that 1 is a quasi-co-2-absorbing (resp. semi-co-2- absorbing) preradical and G is a generator R-module. Since αG G = 1, the result follows from Theorem 3. (2) By Proposition 6 and Theorem 3. Example 1. Let R be a semisimple Artinian ring and S1, S2, . . . , Sn+1 ∈ R-simp be distinct. Then the injective Artinian R-module ⊕n+1 i=1 Si is quasi-co-n-absorbing, by Corollary 7(2). But note that, by [22, Proposi- tion 3.6] and Theorem 2, ⊕n+1 i=1 Si is not co-n-absorbing. This example shows that the two concepts of quasi-co-n-absorbing modules and of co-n-absorbing modules are different in general. The following two propositions can be proved similar to [22, Proposi- tion 4.10] and [22, Theorem 4.11], respectively. Proposition 36. Let N, H ∈ R-co-ass such that H be a fully invariant submodule of N and N be self-injective. For a fully invariant submodule K of H, (1) If K is quasi-co-n-absorbing in N , then K is quasi-co-n-absorbing in H. (2) If K is quasi-co-n-absorbing in N and K ∈ R-co-ass, then K is a quasi-co-n-absorbing module. (3) If αN K is a quasi-co-n-absorbing preradical and N satisfies the ω- property, then αH K is a quasi-co-n-absorbing preradical. 236 Generalizations of semicoprime preradicals Proposition 37. Let N, Q ∈ R-co-ass such that N be a fully invariant submodule of Q and Q be self-injective. Then N is a quasi-co-n-absorbing module if and only if N is quasi-co-n-absorbing in Q. Theorem 4. Let M ∈ R-co-ass that satisfies the ω-property. The follow- ing statements are equivalent: (1) M is quasi-co-n-absorbing; (2) αM M is quasi-co-n-absorbing; (3) For each τ, η ∈ R-pr, M ∈ T(τ[n]:η) ⇒ M ∈ Tτ[n] or M ∈ T(τ[n−1]:η). Proof. (1) ⇔ (2) Is clear by Theorem 3. (2) ⇒ (3) Suppose that αM M is quasi-co-n-absorbing. Let τ, η ∈ R-pr such that M ∈ T(τ[n]:η). Then (τ[n] : η)(M) = M , and so αM M � (τ[n] : η). Therefore αM M � τ[n] or αM M � (τ[n−1] : η). Hence τ[n](M) = M or (τ[n−1] : η)(M) = M . Consequently M ∈ Tτ[n] or M ∈ T(τ[n−1]:η). (3) ⇒ (2) has a routine verification. Similarly we can prove the following theorem. Theorem 5. Let M ∈ R-co-ass that satisfies the ω-property. The follow- ing statements are equivalent: (1) M is semi-co-n-absorbing; (2) αM M is semi-co-n-absorbing; (3) For each τ ∈ R-pr, M ∈ Tτ[n+1] ⇒ M ∈ Tτ[n] . Theorem 6. Let M ∈ R-Mod be such that, for each pair K, L of fully invariant submodules of M , we have ( ωM K : ωM L ) = ωM (K:L). Then, for each quasi-co-n-absorbing (resp. semi-co-n-absorbing) preradical σ such that σ(M) 6= 0, we have that σ(M) is quasi-co-n-absorbing (resp. semi-co-n- absorbing) in M . Proof. By hypothesis M ∈ R-co-ass, [22, Lemma 4.12]. Let σ be a quasi- co-n-absorbing preradical such that σ(M) 6= 0. If K, L are fully invariant submodules of M such that σ(M) 6 (K[n] : L), then σ � ωM σ(M) � ωM (K[n]:L) = ( (ωM K )[n] : ωM L ) . Since σ is quasi-co-n-absorbing, then σ � (ωM K )[n] or σ � ( (ωM K )[n−1] : ωM L ) . 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Mostafanasab Department of Mathematics and Applications, University of Mohaghegh Ardabili, P. O. Box 179, Ardabil, Iran E-Mail(s): yousefian@uma.ac.ir, h.mostafanasab@gmail.com Web-page(s): www.yousefiandarani.com Received by the editors: 21.09.2015 and in final form 27.11.2015.

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