On Generalizations of Graded $r$-ideals

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 $\varphi :GI(R)\to GI(R)\bigcup \{\varnothing \}$ is a function. A proper graded ideal $P$ of $R$ is called a graded $\varphi$-$r$-ideal of $R$ if whenever $x,y$ are homogeneous elements of $R$ such that $xy\in P-\varphi (P)$ and $Ann(x)=\{0\}$, then $y\in P$. Several properties of graded $\varphi$-$r$-ideals have been examined.

References in corpus (1)

• On Generalizations of Graded $2$-absorbing and Graded $2$-absorbing primary submodules