On Graded $\phi$-Prime Submodules

TL;DR

This paper introduces and studies the concept of graded -prime submodules in graded modules over graded rings, generalizing prime and weakly prime submodules, and explores their properties.

Contribution

It defines graded -prime submodules using a function  and investigates their properties, extending the theory of prime submodules.

Findings

Characterization of graded -prime submodules

Relationship with graded prime and weakly prime submodules

Properties and conditions for graded -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 $\phi: GS(M)\rightarrow GS(M)\bigcup\{\emptyset\}$ be a function. A proper graded $R$-submodule $K$ of $M$ is said to be a graded $\phi-$prime $R$-submodule of $M$ if whenever $r$ is a homogeneous element of $R$ and $m$ is a homogeneous element of $M$ such that $rm\in K-\phi(K)$, then either $m\in K$ or $r\in (K:_{R}M)$. If $\phi(K)=\emptyset$ for all $K\in GS(M)$, then a graded $\phi-$prime submodule is exactly a graded prime submodule. If $\phi(K)=\{0\}$ for all $K\in GS(M)$, then a graded $\phi-$prime submodule is exactly a graded weakly prime submodule. Several properties of graded $\phi-$prime submodules have been investigated.