ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2015, Vol. 36, No. 1, pp. 58–64. c  Pleiades Publishing, Ltd., 2015. On 2-Absorbing Submodules over Commutative Rings Manish Kant Dubey 1* and Pakhi Aggarwal 2** 1 SAG, DRDO, Metcalf House, Delhi 110054, India 2 Department of Mathematics, University of Delhi, Delhi 110007, India Abstract—In this paper, we study the concepts of 2-absorbing submodules and weakly 2-absorbing submodules over commutative ring with non-zero identity which are generalizations of prime submodules. Further, we characterized 2-absorbing submodules with flat submodules. DOI: 10.1134/S1995080215010072 Keywords and phrases: 2-absorbing submodules, weakly 2-absorbing submodules, irreducible submodules, flat modules 1. INTRODUCTION Throughout this paper ring R is considered as commutative with a nonzero identity and module M is unital. The prime ideal in ring theory plays crucial role in algebra. Several authors have extended and generalized this concept in several ways. In [3], Anderson and Smith introduced a new concept called weakly prime ideals in a commutative ring R. Further, Badawi [1] and later Badawi and Darani [2] have given new generalization of prime ideals and weakly prime ideals in a commutative ring R in terms of 2- absorbing and weakly 2-absorbing ideals of a ring R. Prime submodules over commutative rings are also always important for the study of module theory. Many researchers have characterized and generalized the notion of prime submodules in a different way. In [6], Atani and Farzalipour were introduced the notion of weakly prime submodules. Recently, Payrovi and Babaei [10] have studied new class of prime submodules say 2-absorbing submodules and further Darani and Soheilnia [4] have also studied the notion of 2-absorbing and weakly 2-absorbing submodules which are obviously generalizations of 2- absorbing and weakly 2-absorbing ideals of commutative rings. Recall [1–4, 6, 10], the following: An ideal I of R is said to be a prime (respectively, weakly prime) if ab ∈ I (respectively 0 = ab ∈ I ) implies that a ∈ I or b ∈ I for all a, b ∈ R. An ideal I of R is said to be proper if I = R. A nonzero proper ideal I of R is said to be a 2-absorbing (respectively, weakly 2-absorbing) ideal of R if abc ∈ I (respectively, 0 = abc ∈ I ) for any a, b, c ∈ R then ab ∈ I or ac ∈ I or bc ∈ I . A proper submodule N of M is said to be a prime (respectively, weakly prime) submodule of M if for a ∈ R and m ∈ M , am ∈ N (respectively, 0 = am ∈ N ) implies that m ∈ N or a ∈ (N : R M ). Let M be an R-module and N a proper submodule of M . Then N is said to be 2-absorbing submodule (respectively, weakly 2-absorbing submodule) of M if whenever a, b ∈ R and m ∈ M with abm ∈ N (respectively, 0 = abm ∈ N ) then ab ∈ (N : R M ) or am ∈ N or bm ∈ N . The annihilator of M denoted by ann R (M ) is (0 : R M ). An R-module M is called a multiplication module if every submodule N of M has the form IM for some ideal I of R. It is easy to see that from [12], N =(N : R M )M because I ⊆ (N : R M ) implies N = IM ⊆ (N : R M )M ⊆ N . If N = IM for some ideal I of R and submodule N of M , then we say that I is a presentation ideal of N . Clearly, every submodule of M has a presentation ideal if and only if M is a multiplication module. Let N and K be submodules of a multiplication R- module M with N = I 1 M and K = I 2 M for some ideals I 1 and I 2 of R. The product of N and K denoted by NK, is defined by NK = I 1 I 2 M . By [7], we have the product of N and K is independent of presentations of N and K. Also, the term ab for some a, b ∈ M represents the product of Ra and * E-mail: kantmanish@yahoo.com ** E-mail: scarlet2k\_81@yahoo.com 58