Generalizations of prime submodules over non-commutative rings

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 $\phi:S(M)\rightarrow S(M)\cup{\emptyset}$isafunction\mathrm{.}Forevery$Y\in S(M)$andideal$I$of$R,$apropersubmodule$X$of$M$iscalled$\phi$-prime,if$YI\subseteq X$and$YI\nsubseteq\phi(X),$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.