ON 2-ABSORBING PRIMARY SUBMODULES OF MODULES OVER COMMUTATIVE RINGS

Abstract

All rings are commutative with 1 not equal 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2 -absorbing primary submodules generalizing 2 -absorbing primary ideals of rings. Let M be an R -module. A proper submodule N of an R -module M is called a 2 -absorbing primary submodule of M if whenever a, b is an element of R and m is an element of M and abm is an element of N, then am is an element of M-rad(N) or bm is an element of M-rad(N) or ab is an element of (N :(R) M). It is shown that a proper submodule N of M is a 2 -absorbing primary submodule if and only if whenever I-1 I-2 K subset of N for some ideals I-1,I-2 of R and some submodule K of M, then I-1,I-2 subset of (N :(R) M) or I-1 K subset of M-rad(N) or I2K subset of M-rad(N). We prove that for a submodule N of an R -module M if M-rad(N) is a prime submodule of M, then N is a 2 -absorbing primary submodule of M. If N is a 2 -absorbing primary submodule of a finitely generated multiplication R -module M, then (N :(R) M) is a 2 -absorbing primary ideal of R and M-rad(N) is a 2 -absorbing submodule of M.