On graded primary-like submodules of graded modules over graded commutative rings

TL;DR

This paper introduces the concept of graded primary-like submodules in graded modules over graded commutative rings, generalizing graded primary ideals, and explores their properties, especially when satisfying the gr-primeful property.

Contribution

It proposes a new class of submodules called graded primary-like submodules and investigates their fundamental properties and special cases.

Findings

Defined graded primary-like submodules as a generalization of graded primary ideals.

Established basic properties of these submodules.

Analyzed properties when submodules satisfy the gr-primeful property.

Abstract

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to find extra properties of these graded submodules.