On Generalizations of Graded $r$-ideals
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.