On Generalizations of Graded $r$-ideals

TL;DR

This paper introduces a broad generalization of graded r-ideals in graded commutative rings, exploring their properties and extending the theoretical framework of ideal theory in algebra.

Contribution

It defines graded φ-r-ideals in graded rings and investigates their properties, expanding the concept of graded r-ideals in algebraic structures.

Findings

Defined graded φ-r-ideals and established their fundamental properties.

Extended the theory of graded r-ideals to a more general setting.

Provided conditions under which graded φ-r-ideals exhibit specific behaviors.

Abstract

In this article, we introduce a generalization of the concept of graded $r$-ideals in graded commutative rings with nonzero unity. Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all graded ideals of $R$. Suppose that $\phi: GI(R)\rightarrow GI(R)\bigcup\{\emptyset\}$ is a function. A proper graded ideal $P$ of $R$ is called a graded $\phi$-$r$-ideal of $R$ if whenever $x, y$ are homogeneous elements of $R$ such that $xy\in P-\phi(P)$ and $Ann(x) =\{0\}$, then $y\in P$. Several properties of graded $\phi$-$r$-ideals have been examined.