r. abu dawwas

Let $G$ be a group with identity $e$ and $R$ be a commutative $G$-graded ring with nonzero unity $1$. In this article, we introduce the concept of graded $r$-ideals. A proper graded ideal $P$ of a graded ring $R$ is said to be graded $r$-ideal if whenever $a, bin h(R)$ such that $abin P$ and $Ann(a)={0}$, then $bin P$. We study and investigate the behavior of graded $r$-ideals to introduce  several results. We introduced several characterizations for graded $r$-ideals;  we proved that $P$ is a graded $r$-ideal of $R$ if and only if $aP=aRbigcap P$  for all $ain h(R)$ with $Ann(a)={0}$. Also, $P$ is a graded $r$-ideal of $R$  if and only if $P=(P:a)$ for all $ain h(R)$ with $Ann(a)={0}$. Moreover,  $P$ is a graded $r$-ideal of $R$ if and only if whenever $A, B$ are graded ideals of   $R$ such that $ABsubseteq P$ and $Abigcap r(h(R))neqphi$, then $Bsubseteq P$.    In this article, we introduce the concept of $huz$-rings. A graded ring $R$    is said to be $huz$-ring if every homogeneous element of $R$ is either a zero       if every graded ideal of $R$ is a graded $r$-ideal. Moreover, assuming that  $R$ is a graded domain, we proved that ${0}$ is the only graded $r$-ideal of $R$.