S-1-absorbing primary submodules

Abstract

In this work, we introduce the notion of S-1-absorbing primary submodule as an extension of 1-absorbing primary submodule. Let S be a multiplicatively closed subset of a ring R and M be an R-module. A submodule N of M with (N : R M )∩S = ∅ is said to be S-1-absorbing primary if whenever abm ∈ N for some non-unit a, b ∈ R and m ∈ M , then either sab ∈ (N : R M ) or sm ∈ M -rad(N ). We examine several properties of this concept and provide some characterizations. In addition, S-1-absorbing primary avoidance theorem and S-1-absorbing primary property for idealization and amalgamation are presented.