Modules with ⊛(⊛’or ⊛’’) Condition

In this paper we introduce and study modules with ⊛ (⊛ or ⊛) condition. We give several properties of these types of modules and some relationships between them. Introduction Let M be an R-module, where R is a commutative ring with unity. Recall that a nonzero submodule N of M is called strongly hollow (briefly, SH-submodule) if whenever L1, L2  M, N  L1 + L2 implies either N  L1 or N  L2, [1]. A submodule N of M is called strongly irreducible (SIsubmodule) if whenever L1, L2  M, N  L1  L2 implies N  L1 or N  L2, [4]. The sets {K:K is a SH-submodule of M} {K:K is a proper SH-submodule of M} and {K:K is a nonzero proper SIsubmodule of M} are denoted by respectively,[8].In [8] we studied and topologized these sets by setting that for any L ≤ M We prove that is a topological space (see [8, Th.2.1.9)]. Also we see that is not closed under finite union, however all other axioms of closed sets of a topological space are valid (see [8, Th.2.4.1]). This leaded us to call an R-module M a -module if is closed under finite union. Equivalently M is a -module if is a topological space. Beside these we see that is not closed under finite union, however all other axioms of closed sets of a topological space are valid (see [8, Th.3.2.1]). This leaded us to call an R-module M is a module if is closed under a finite union. Equivalently M is a module if is a topological space. We notice that, for any L1  M, L2  M, if then it is not necessarily that L1 = L2, as the following examples show. (1) Consider the Z-module Z, but 3Z  {0}. (2) For the Z-module Z12, (3) For the Z-module Z12, but  6   {0} . The Bulletin of Society for Mathematical Services and Standards Online: 2012-12-03 ISSN: 2277-8020, Vol. 4, pp 12-21 doi:10.18052/www.scipress.com/BSMaSS.4.12 © 2012 SciPress Ltd., Switzerland SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/ These observations lead us to introduce the following conditions: This paper is devoted to study modules with ⊛ (⊛, ⊛ respectively). Also we shall study the behaviour respectively when M satisfies ⊛ (⊛, ⊛). S.1 Modules with the Condition ⊛ We start this by the following remarks and examples. Remarks and Examples 1.1: (1) The Z-module Z does not satisfies ⊛ since for each L, N  Z, L  Z, (2) Every simple module satisfies ⊛. (3) Let M, M be two isomorphic R-modules. Then M satisfies ⊛ if and only if M satisfies ⊛. (4) Let M1, M2 be R-modules. If M1, M2 satisfies ⊛ condition, then M1  M2 may not be satisfy ⊛, as an example: Let M1 = Z3 as a Z-module and M2 = Z4 as a Z-module. Each of M1 and M2 satisfies ⊛. However Z3  Z4 ≅ Z12 and Z12 does not satisfy ⊛. (5) Let M be an R-module. Then M satisfies ⊛ as R-module if and only if M satisfies ⊛ asR –module where R = R/ann M. Proposition 1.2. Let M be an R-module such that every nonzero submodules is SH. Then M satisfies ⊛. Proof: First note that for each N  <0>. Let L, N  M, L  (0), N  (0) such that . Since L  L and L is a SH-submodule by hypothesis, . It follows that L  N. Similarly, and hence N  L. Thus L = N. Recall that an R-module M is called chained if the lattice of its submodules is linearly ordered by inclusion [10]. Corollary 1.3. Let M be a chained R-module. Then M satisfies ⊛. The following theorem gives a characterization of modules with the condition ⊛. Theorem 1.4: Let M be an nonzero R-module. Then M satisfies ⊛ if and only if every nonzero submodule of M can be represented as sum of SH-submodules. Proof: () Let (0)  K  M. Then and hence K = (0) (by ⊛), which is a contradiction. Set W and let L V (N) . Then L  W V (K) But for each W  V (K) , W  K, so N  K. Thus L  K and hence L  Thus  The Bulletin of Society for Mathematical Services and Standards Vol. 4 13


Introduction
Let M be an R-module, where R is a commutative ring with unity. Recall that a nonzero submodule N of M is called strongly hollow (briefly, SH-submodule) if whenever L 1 , L 2  M, N  L 1 + L 2 implies either N  L 1 or N  L 2 , [1]. A submodule N of M is called strongly irreducible (SIsubmodule) if whenever L 1 , L 2  M, N  L 1  L 2 implies N  L 1 or N  L 2 , [4]. The sets {K:K is a SH-submodule of M} {K:K is a proper SH-submodule of M} and {K:K is a nonzero proper SIsubmodule of M} are denoted by respectively, [8].In [8] we studied and topologized these sets by setting that for any L ≤ M We prove that is a topological space (see [8,Th.2.1.9)]. Also we see that is not closed under finite union, however all other axioms of closed sets of a topological space are valid (see [8,Th.2.4.1]). This leaded us to call an R-module M a -module if is closed under finite union. Equivalently M is a -module if is a topological space.
Beside these we see that is not closed under finite union, however all other axioms of closed sets of a topological space are valid (see [8,Th.3.2.1]). This leaded us to call an R-module M is a module if is closed under a finite union. Equivalently M is a module if is a topological space.
We notice that, for any L 1  M, L 2  M, if then it is not necessarily that L 1 = L 2 , as the following examples show.
where R = R/ann M. Recall that a submodule N of an R-module is called second if for each ideal I of R, either IK = K or IK = (0), [13].
To give our next result, first we introduce the following Lemma.

S.2 Modules with the Condition ⊛'
In this section, we study modules that satisfy ⊛. Some properties of these modules are analogus to that of modules with the condition ⊛.
As we mention in the introduction, a module with condition ⊛ if it satisfies the condition ⊛, where 16 Volume 4
(2) If every proper nonzero submodule of M is SH, then M satisfies ⊛ for each proper nonzero submodules of M.  (1) and (2), and by ⊛, K  N= , i.e. K is an intersection of SHsubmodules
To give the next result we need the following lemmas. First compare the first lemma with lemma 1.9.

Lemma 2.7. Let M be an R-module. Then is a T 1 -space if and only if every proper SHsubmodule is maximal SH in Spec(M) .
Proof: It is analogus to the proof of lemma 1.9, so is omitted. Recall that a topological space (X,) is called cofinite if the only closed subsets of X are finite sets or X. Equivalently  = {U: U  X and X -U is a finite set}  {}.

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Volume 4 Theorem 2.10. Let M be a faithful finitely generated multiplication over a comultiplication ring.
Then is a T 1 -space and M satisfies ⊛ if and only if M is cosemisimple and every proper SH-submodule of M is a maximal SH of M. Proof: () By lemma 2.7, every proper SH-submodule is a maximal SH-submodule and by Lemma 2.9, every maximal SH-submodule is a maximal submodule. Thus every proper SHsubmodule is a maximal submodule. Moreover by Th.2.2, every proper submodule of M is an intersection of SH-submodule. Thus every proper submodule is an intersection of maximal submodules of M; that is M is cosemisimple.
() By lemma 2.7, is T 1 . But M is cosemiple, so every submodule is an intersection of maximal submodules. Hence by Lemma 2.9, every proper submodule is an intersection of SHsubmodules. Hence by Th.2.2, M satisfies⊛. Compare the following result with Th.1.14. Theorem 2.11. Let M be a finitely generated faithful multiplication R-module. Then M satisfies ⊛if and only if R satisfies ⊛. Proof: It is similar to the proof of Th.1.14, so is omitted. Remark 2.12. The condition "M is faithful" is necessary in Th.2.11, as for example: The Z-module Z 6 is a finitely generated not faithful multiplication Z-module and satisfies ⊛. But the ring Z does not satisfies ⊛, since for any I, J  R. Corollary 2.13. Let M be a finitely generated multiplication R-module. Then the following statements are equivalent: (1) M satisfies ⊛ as R-module.
Recall that a proper submodule N of an R-module M is called prime if whenever r  R, x  M, rx  N implies x  N or r  [N:M], [9]. Equivalently a proper submodule N of an R-module M is prime if for any ideal I of R and for any K  M, IK  N implies K  N or I  [N:M], [9]. Compare the following Lemma with Lemma 1.12. Lemma 2.14. Let M be an R-module such that every proper SH-submodule is prime. Then for any I

S.3 Modules with the Condition ⊛
In this section, we introduce modules that satisfy ⊛, where ⊛ : for each N, L  M, implies N = L.
Many results about these modules are similar to that of module with ⊛ condition. Also we give some relations modules with ⊛ and modules with condition ⊛. Remark and Examples 3.1 (1) Every simple module M does not satisfy ⊛, since but M  <0>.
The following theorem is similar to Th. The following Lemma is similar to Lemma 1.11. Lemma 3.5. Let M be a faithful finitely generated multiplication over comultiplication ring R, let N  M. If N is a minimal SI-submodule, then N is simple.