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) [6].

The notion of 2-absorbing ideals as a generalization of prime ideals was introduced and studied in [3]. A proper ideal I of 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. The authors in [5] and [12], extended 2-absorbing ideals to 2-absorbing submodules. A proper submodule N of an R-module 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).

The purpose of this paper is to introduce the concepts of quasi 2-absorbing submodules as a generalization of 2-absorbing submodules and obtain some related results.

Definition 2.1

We say that a proper submodule N of an R-module M is a quasi 2-absorbing submodule if (N :_R M) is a 2-absorbing ideal of R.

Example 2.2

By [12, 2.3], every 2-absorbing submodule is a quasi 2-absorbing submodule. But the converse is not true in general. For example, the submodules 〈1/p² + Z〉 and 〈1/p + Z〉 of the Z-module Z_p∞ are quasi 2-absorbing submodules which are not 2-absorbing submodules.

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 [4].

Proposition 2.3

Let M be a multiplication R-module. Then a submodule N of M is a 2-absorbing submodule of M if and only if it is a quasi 2-absorbing submodule of M.

Proposition 2.4

Let M be an R-module and N₁, N₂be two submodules of M with (N₁ :_R M) and (N₂ :_R M) prime ideals of R. Then N₁ ∩ N₂is a quasi 2-absorbing submodule of M.

Theorem 2.5

Let M be an R-module and N be a quasi 2-absorbing submodule of M. Then we have the following:

• (a) (N :_M I) is a quasi 2-absorbing submodules of M for all ideals I of R with I ⊈ (N :_R M).

• (b) If I is an ideal of R, then (N :_R IⁿM) = (N :_R Iⁿ⁺¹M), for all n ≥ 2.

• (c) If M is a finitely generated Laskerian R-module, then rad(N) is a quasi 2- absorbing submodule of M.

Corollary 2.6

Let M be a comultiplication R-module such that the zero submodule of M is a quasi 2-absorbing submodule. Then every proper submodule of M is a quasi 2-absorbing submodule of M.

Proposition 2.7

Let M be an R-module and {K_i}_i∈I be a chain of quasi 2- absorbing submodules of M. Then ∩_i∈IK_i is a quasi 2-absorbing submodule of M.

Definition 2.8

We say that a quasi 2-absorbing submodule N of an R-module M is a minimal quasi 2-absorbing submodule of a submodule K of M, if K ⊆N and there does not exist a quasi 2-absorbing submodule T of M such that K ⊂ T ⊂ N.

It should be noted that a minimal quasi 2-absorbing submodule of M means that a minimal quasi 2-absorbing submodule of the submodule 0 of M.

Lemma 2.9

Let M be an R-module. Then every quasi 2-absorbing submodule of M contains a minimal quasi 2-absorbing submodule of M.

Theorem 2.10

Let M be a Noetherian R-module. Then M contains a finite number of minimal quasi 2-absorbing submodules.

Proposition 2.11

Let N be a submodule of a finitely generated R-module M and S be a multiplicatively closed subset of R. If N is a quasi 2-absorbing submodule and (N :_R M) ∩ S = ∅︀, then S⁻¹N is a quasi 2-absorbing S⁻¹R-submodule of S⁻¹M. Furthermore, if S⁻¹N is a quasi 2-absorbing S⁻¹R-submodule and S ∩ Z(R/(N :_R M)) = ∅︀, then N is a quasi 2-absorbing submodule of M.

Lemma 2.12

Let f : M → Ḿ be a monomorphism of R-modules. Then N is a quasi 2-absorbing submodule of M if and only if f(N) is a quasi 2-absorbing submodule of f(M).

Lemma 2.13. ([7, Corollary 2.11])

Let N be a submodule of a multiplication Rmodule M. Then N is a prime submodule of M if and only if (N :_R M) is a prime ideal of R.

Let R_i be a commutative ring with identity and M_i be an R_i-module for i = 1, 2. Let R = R₁ × R₂. Then M = M₁ ×M₂ is an R-module and each submodule of M is in the form of N = N₁ × N₂ for some submodules N₁ of M₁ and N₂ of M₂.

Theorem 2.14

Let R = R₁×R₂be a decomposable ring and let M = M₁ × M₂be an R-module, where M₁is a multiplication R₁-module and M₂is a multiplication R₂-module. Suppose that N = N₁ × N₂is a proper submodule of M. Then the following conditions are equivalent:

• (a) N is a quasi 2-absorbing submodule of M;

• (b) Either N₁ = M₁and N₂is a quasi 2-absorbing submodule of M₂or N₂ = M₂and N₁is a quasi 2-absorbing submodule of M₁or N₁, N₂are prime submodules of M₁, M₂, respectively.

Theorem 2.15

Let R = R₁ × R₂ × ··· ×R_n (2 ≤ n < ∞) be a decomposable ring and M = M₁ × M₂ ··· × M_n be an R-module, where for every 1 ≤ i ≤ n, M_i is a multiplication R_i-module, respectively. Then for a proper submodule N of M the following conditions are equivalent:

• (a) N is a quasi 2-absorbing submodule of M;

• (b) EitherN=×i=1nNisuch that for some k ∈ {1, 2, …, n}, N_k is a quasi 2- absorbing submodule of M_k and N_i = M_i for every i ∈ {1, 2, …, n} {k} orN=×i=1nNisuch that for some k,m ∈ {1, 2, …, n}, N_k is a prime submodule of M_k, N_m is a prime submodule of M_m, and N_i = M_i for every i ∈ {1, 2, …, n} {k,m}.

The author would like to thank Prof. Habibollah Ansari-Toroghy for his helpful suggestions and useful comments.