Generalizations of 2-absorbing and 2-absorbing primary submodules

Abstract

In this study, we introduce phi-2-absorbing and phi-2-absorbing primary submodules of modules over commutative rings generalizing the concepts of 2-absorbing and 2-absorbing primary submodules. Let phi : S(M) -> S(M) boolean OR {phi} be a function where S(M) denotes the set of all submodules of M and N a proper submodule of an R-module M. We will say that N is a phi-2-absorbing submodule of M if whenever a, b is an element of R, m is an element of M with abm is an element of N and abm (sic) phi(N), then am is an element of N or bm is an element of N or ab is an element of (N :(R) M) and N is said to be a phi-2-absorbing primary submodule of M whenever if a, b is an element of R, m is an element of M with abm is an element of N and abm (sic) phi(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). We investigate many properties of these new types of submodules and establish some characterizations for phi-2-absorbing and phi-2-absorbing primary submodules of multiplication modules.