Generalizations of prime submodules over non-commutative rings

Throughout this paper, R is an associative ring (not necessarily cmmutative) with identity and M is a right R-module with unitary. In aper, we introduce a new concept of φ-prime submodule overan 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\} is a function. For every Y\in S(M) and ideal I of R, a proper submodule X of M is called \textit{\phi-prime,} if YI\subseteq X and YI\nsubseteq\phi(X), then Y\subseteq X or I\subseteq(X:_{R}M)\mathit{.} Then we examine the properties of \textit{\phi-}prime submodules and characterize it when M is a \textit{multiplication module.}

Keywords:

\phi-prime Submodule , Non-commutative Ring , Multiplication Module