On 2-absorbing submodule elements in le-modules and its generalizations

Abstract

In this paper, we introduce the concept of 2-absorbing submodule elements in an le-module M as follows: a proper submodule element q in M is said to be 2-absorbing for any r,s is an element of R and m is an element of M if rsm <= q, then either rs is an element of (q : e) or rm <= q or sm <= q. Moreover, we define some generalizations of the new concept such as weakly 2-absorbing, n-absorbing, weakly n-absorbing, (n,k)-absorbing, weakly (n,k)-absorbing submodule elements in le-modules. After presenting a main example for le-modules, we study some counter examples for the generalizations. In addition to giving some characterizations for the new concepts, we investigate the relationship between prime (primary) submodule elements and them.