Generalizations of prime submodules over non-commutative rings

Abstract

Throughout this paper, $R$ is an associative ring (not necessarily commutative) with identity and $M$ is a right $R$-module with unitary. In this paper, we introduce a new concept of $\varphi$-prime submodule over an associative ring with identity. Thus we define the concept as following: Assume that $S(M)$ is the set of all submodules of $M$ and $\varphi :S(M)\to S(M)\cup \{\varnothing \}$ is a function. For every $Y\in S(M)$ and ideal $I$ of $R,$ a proper submodule $X$ of $M$ is called $\varphi$-prime, if $YI\subseteq X$ and $YI⊈\varphi (X),$ then $Y\subseteq X$ or $I\subseteq (X{:}_{R}M)$. Then we examine the properties of $\varphi$-prime submodules and characterize it when $M$ is a multiplication module.