Classical and strongly classical 2-absorbing second submodules

In this paper, we will introduce the concept of classical (resp. strongly classical) 2-absorbing second submodules of modules over a commutative ring as a generalization of 2-absorbing (resp. strongly 2-absorbing) second submodules and investigate some basic properties of these classes of modules.


Introduction
Throughout this paper, R will denote a commutative ring with identity and "⊂" will denote the strict inclusion. Further, Z will denote the ring of integers.
Let M be an R-module. A proper submodule P of M is said to be prime if for any r ∈ R and m ∈ M with rm ∈ P , we have m ∈ P or r ∈ (P : R M ) [11]. Let N be a submodule of M . A non-zero submodule S of M is said to be second if for each a ∈ R, the homomorphism S a → S is either surjective or zero [18]. In this case Ann R (S) is a prime ideal of R.
The notion of 2-absorbing ideals as a generalization of prime ideals was introduced and studied in [7]. A proper ideal I of R is a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. The authors in [10] and [15], extended 2-absorbing ideals to 2-absorbing submodules. A proper submodule N of M is called a 2-absorbing submodule of M if whenever abm ∈ N for some a, b ∈ R and m ∈ M , then am ∈ N or bm ∈ N or ab ∈ (N : R M ).
In [5], the present authors introduced the dual notion of 2-absorbing submodules (that is, 2-absorbing (resp. strongly 2-absorbing) second submodules) of M and investigated some properties of these classes of modules. A non-zero submodule N of M is said to be a 2-absorbing second submodule of M if whenever a, b ∈ R, L is a completely irreducible submodule of M , and abN ⊆ L, then aN ⊆ L or bN ⊆ L or ab ∈ Ann R (N ). A non-zero submodule N of M is said to be a strongly 2-absorbing second submodule of M if whenever a, b ∈ R, K is a submodule of M , and abN ⊆ K, then aN ⊆ K or bN ⊆ K or ab ∈ Ann R (N ).
In [14], the authors introduced the notion of classical 2-absorbing submodules as a generalization of 2-absorbing submodules and studied some properties of this class of modules. A proper submodule N of M is called classical 2-absorbing submodule if whenever a, b, c ∈ R and m ∈ M with abcm ∈ N , then abm ∈ N or acm ∈ N or bcm ∈ N [14].
The purpose of this paper is to introduce the concepts of classical and strongly classical 2-absorbing second submodules of an R-module M as dual notion of classical 2-absorbing submodules and provide some information concerning these new classes of modules. We characterize classical (resp. strongly classical) 2-absorbing second submodules in Theorem 2.3 (resp. Theorem 3.4). Also, we consider the relationship between classical 2-absorbing and strongly classical 2-absorbing second submodules in Examples 3.9, 3.10, and Propositions 3.11. Theorem 2.14 (resp. Theorem 3.15) of this paper shows that if M is an Artinian R-module, then every non-zero submodule of M has only a finite number of maximal classical (resp. strongly classical) 2-absorbing second submodules. Further, among other results, we investigate strongly classical 2-absorbing second submodules of a finite direct product of modules in Theorem 3.19.

Classical 2-absorbing second submodules
It is easy to see that every submodule of M is an intersection of completely irreducible submodules of M [12].
We frequently use the following basic fact without further comment.
Remark 2.1. Let N and K be two submodules of an R-module M . To prove      Proof. (a) It is enough to show that a 2 N = a 3 N . It is clear that a 3 N ⊆ a 2 N . Let L be a completely irreducible submodule of M such that a 3 N ⊆ L. Since N is a classical 2-absorbing second submodule, a 2 N ⊆ L. This implies that a 2 N ⊆ a 3 N . (b) Assume that a, b, c ∈ R and abc ∈ (L : R N ). Then there is a positive integer t such that a t b t c t N ⊆ L. By hypotheses, N is a classical 2-absorbing second ( By [5, 3.14], f (L) is a completely irreducible submodule of f (M ). Thus asŃ is a classical 2-absorbing second submodule, An R-module M is said to be a multiplication module if for every submodule N of M there exists an ideal I of R such that N = IM [8].
An R-module M is said to be a comultiplication module if for every submodule N of M there exists an ideal I of R such that N = (0 : M I), equivalently, for each submodule N of M , we have N = (0 : M Ann R (N )) [2]. Proof. This follows from parts (a) and (b) of Lemma 2.9.
Proposition 2.11. Let M be an R-module and {K i } i∈I be a chain of classical 2absorbing second submodules of M . Then i∈I K i is a classical 2-absorbing second submodule of M .
Proof. Let a, b, c ∈ R, L be a completely irreducible submodule of M , and abc i∈I K i ⊆ L. Assume that ab i∈I K i ⊆ L and ac i∈I K i ⊆ L. Then there are m, n ∈ I where abK n ⊆ L and acK m ⊆ L. Hence, for every K n ⊆ K s and every K m ⊆ K d we have that abK s ⊆ L and acK d ⊆ L. Therefore, for each submodule K h such that K n ⊆ K h and K m ⊆ K h , we have bcK h ⊆ L. Hence bc i∈I K i ⊆ L, as needed. Proof. This is proved easily by using Zorn's Lemma and Proposition 2.11.
We say M is a strongly classical 2-absorbing second module if M is a strongly classical 2-absorbing second submodule of itself.
Clearly every strongly classical 2-absorbing second submodule is a classical 2absorbing second submodule.    Proof. It is enough to show that I 2 N = I 3 N . By Theorem 3.4, I 2 N = I 3 N . Example 3.6. Clearly every strongly 2-absorbing second submodule is a strongly classical 2-absorbing second submodule. But the converse is not true in general. For example, consider M = Z 6 ⊕ Q as a Z-module. Then M is a strongly classical 2-absorbing second module. But M is not a strongly 2-absorbing second module.
A non-zero submodule N of an R-module M is said to be a weakly second submodule of M if rsN ⊆ K, where r, s ∈ R and K is a submodule of M , implies either rN ⊆ K or sN ⊆ K [1]. For a submodule N of an R-module M the the second radical (or second socle) of N is defined as the sum of all second submodules of M contained in N and it is denoted by sec(N ) (or soc(N )). In case N does not contain any second submodule, the second radical of N is defined to be (0) (see [9] and [3]). The following examples show that the two concepts of classical 2-absorbing submodules and strongly classical 2-absorbing second submodules are different in general.
Example 3.9. The submodule 2Z of the Z-module Z is a classical 2-absorbing submodule which is not a strongly classical 2-absorbing second module.
Example 3.10. The submodule 1/p + Z of the Z-module Z p ∞ is a a strongly classical 2-absorbing second module which is not a classical 2-absorbing submodule of Z p ∞ .
A commutative ring R is said to be a u-ring provided R has the property that an ideal contained in a finite union of ideals must be contained in one of those ideals; and a um-ring is a ring R with the property that an R-module which is equal to a finite union of submodules must be equal to one of them [16].
In the following proposition, we investigate the relationships between strongly classical 2-absorbing second submodules and classical 2-absorbing submodules. It follows that abM ⊆ abcM . Thus abM = abcM because the reverse implication is clear and this completed the proof.  Proof. This is proved easily by using Zorn's Lemma and Proposition 3.12. Proof. Use the technique of Theorem 2.14 any apply Lemma 3.14.
Thus asŃ is a strongly classical 2-absorbing second submodule, Let R i be a commutative ring with identity and M i be an R i -module for i = 1, 2.   Thus (a, 1) (b) ⇒ (a). Suppose that N = N 1 × 0, where N 1 is a strongly classical 2absorbing (resp. weakly) second submodule of M 1 . Then it is clear that N is a strongly classical 2-absorbing (resp. weakly) second submodule of M . Now, assume that N = N 1 × N 2 , where N 1 and N 2 are weakly second submodules of M 1 and M 2 , respectively. Hence (N 1 × 0) + (0 × N 2 ) = N 1 × N 2 = N is a strongly classical 2-absorbing second submodule of M , by Preposition 3.7 (b).
Lemma 3.18. Let R = R 1 × R 2 × · · · × R n be a decomposable ring and M = M 1 × M 2 · · · × M n be an R-module where for every 1 ≤ i ≤ n, M i is an R i -module, respectively. A non-zero submodule N of M is a weakly second submodule of M if and only if N = × n i=1 N i such that for some k ∈ {1, 2, ..., n}, N k is a weakly second submodule of M k , and N i = 0 for every i ∈ {1, 2, ..., n} \ {k}.
(⇐) This is clear. Proof. We use induction on n. For n = 2 the result holds by Theorem 3.17. Now let 3 ≤ n < ∞ and suppose that the result is valid when K = M 1 × · · · × M n−1 . We show that the result holds when M = K × M n . By Theorem 3.17, N is a strongly classical 2-absorbing second submodule of M if and only if either N = L × 0 for some strongly classical 2-absorbing second submodule L of K or N = 0 × L n for some strongly classical 2-absorbing second submodule L n of M n or N = L × L n for some weakly second submodule L of K and some weakly second submodule L n of M n . Note that by Lemma 3.18, a non-zero submodule L of K is a weakly second submodule of K if and only if L = × n−1 i=1 N i such that for some k ∈ {1, 2, ..., n−1}, N k is a weakly second submodule of M k and N i = 0 for every i ∈ {1, 2, ..., n − 1} \ {k}. Hence the claim is proved.
Example 3.20. Let R be a Noetherian ring and let E = ⊕ m∈Max(R) E(R/m). Then for each 2-absorbing ideal P of R, (0 : E P ) is a strongly classical 2-absorbing second submodule of E.