Square-difference factor absorbing submodules of modules over commutative rings

Abstract

Let R be a commutative ring with identity and M an unitary R-module. Recently, in [5], Anderson, Badawi and Coykendalla defined a proper ideal I of R to be a square-difference factor absorbing ideal (sdf-absorbing ideal) of R if whenever a2 − b2 ∈ I for 0 ≠ a, b ∈ R, then a + b ∈ I or a − b ∈ I. Generally, this article is devoted to introduce and study square-difference factor absorbing submodules. A proper submodule N of M is called square-difference factor absorbing (sdf-absorbing) in M if whenever m ∈ M and a, b ∈ R\AnnR(m) such that (a2 − b2)m ∈ N, then (a + b)m ∈ N or (a − b)m ∈ N. Many properties, examples and characterizations of sdf-absorbing submodules are introduced, especially in multiplication modules. Comparing this new class of submodules with classical prime submodules, we present new characterizations for von-Neumann regular modules in terms of sdf-absorbing submodules. Further characterizations of some special modules in which every nonzero proper submodule is sdf-absorbing are investigated. Finally, the sdf-absorbing submodules in amalgamated modules are studied.