Pseudo 2-absorbing and pseudo semi 2-absorbing submodules

Abstract

All rings in this article are commutative rings with identity, and all R-modules are left unitary R-modules. A proper submodule E of an R-module X is called 2-absorbing if whenever abx ∈ E, where a,b ∈ R, x ∈ X implies that either axx ∈ E ax ∈ E or ab bx ∈ or ab[E: X]. In this paper we introduced and studied the concepts of pseudo 2-absorbing and pseudo semi-2-absorbing submodules as generalizations of 2-absorbing submodule, where a proper submodule E of an R-module X is called pseudo 2-absorbing if whenever abx ∈ E, where a,b ∈ R, x ∈ X implies that either ax∈(E + soc(X)) or bx ∈ (E+soc(X)) or abX ⊆ (E + soc(X)) and a proper submodule E of an R-module X is called pseudo semi-2-absorbing if whenever a2x ∈ E, where a ∈ R, x ∈ X implies that either ax ∈ (E + soc(X)) or a2X ⊆ (E + soc(X)). Many basic Properties, characterizations and examples of these concepts are introduced. Moreover, the relation between these concepts is given. Furthermore, the behavior of these concepts in some types of modules are studied.