On Graded $\varphi$-Prime Submodules

Abstract

Let $R$ be a graded commutative ring with non-zero unity $1$ and $M$ be a graded unitary $R$-module. Let $GS(M)$ be the set of all graded $R$-submodules of $M$ and $\varphi :GS(M)\to GS(M)\bigcup \{\varnothing \}$ be a function. A proper graded $R$-submodule $K$ of $M$ is said to be a graded $\varphi -$prime $R$-submodule of $M$ if whenever $r$ is a homogeneous element of $R$ and $m$ is a homogeneous element of $M$ such that $rm\in K-\varphi (K)$, then either $m\in K$ or $r\in (K{:}_{R}M)$. If $\varphi (K)=\varnothing$ for all $K\in GS(M)$, then a graded $\varphi -$prime submodule is exactly a graded prime submodule. If $\varphi (K)=\{0\}$ for all $K\in GS(M)$, then a graded $\varphi -$prime submodule is exactly a graded weakly prime submodule. Several properties of graded $\varphi -$prime submodules have been investigated.

arXiv admin note: text overlap with arXiv:2101.11151