On Graded $\phi$-$1$-absorbing prime ideals

TL;DR

This paper introduces and explores the properties of graded $\,\phi$-1-absorbing prime ideals in $G$-graded commutative rings, generalizing prime ideal concepts with a focus on their algebraic structure.

Contribution

It defines the new concept of graded $\,\phi$-1-absorbing prime ideals and investigates their fundamental properties within $G$-graded rings.

Findings

Characterization of graded $\,\phi$-1-absorbing prime ideals

Conditions under which these ideals are prime or primary

Examples illustrating the new concept

Abstract

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 $\phi:GI(R)\rightarrow GI(R)\cup\{\emptyset\}$ is a function. In this article, we introduce and study the concept of graded $\phi$-$1$-absorbing prime ideals. A proper graded ideal $I$ of $R$ is called a graded $\phi$% -$1$-absorbing prime ideal of $R$ if whenever $a,b,c$ are homogeneous nonunit elements of $R$ such that $abc\in I-\phi(I)$, then $ab\in I$ or $c\in I$. Several properties of graded $\phi$-$1$-absorbing prime ideals have been examined.