Square-difference factor absorbing submodules of modules over commutative rings

Let R be a commutative ring with identity and M an unitary R-module. Recently, in Bad1, 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 a?-b??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\Ann_{R}(m) such that (a?-b?)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.