On Properties of Graded Rings and Graded Modules

TL;DR

This paper introduces new concepts in graded ring theory, including weakly graded rings, invertible graded rings, and graded semi-uniform modules, exploring their properties and relationships within the structure of graded rings.

Contribution

It presents novel definitions and relationships for weakly graded rings, invertible graded rings, weakly crossed products, and graded semi-uniform modules, expanding the theoretical framework.

Findings

Defined weakly graded rings and invertible graded rings.

Explored properties and relationships of weakly crossed products.

Generalized graded essential submodules and introduced graded semi-uniform modules.

Abstract

Let $R$ be a $G$-graded ring. In this article, we introduce two new concepts on graded rings, namely, weakly graded rings and invertible graded rings, and we discuss the relations between these concepts and several properties of graded rings. Also, we study the concept of weakly crossed products and study some properties defined on weakly crossed product and give the relationship between this new concept and several properties of graded rings. Moreover, in this article, we give a generalization for the concept of graded essential submodules, and introduce the concept of graded semi-uniform modules which is a generalization for the concept of graded uniform modules.