%0 Journal Article
%A Bataineh, M.
%A Abu-dawwas, R.
%T Graded r-Ideals
%J Iranian Journal of Mathematical Sciences and Informatics
%V 14
%N 2
%U http://ijmsi.ir/article-1-984-en.html
%R
%D 2019
%K Graded prime ideals, Graded r-ideals.,
%X 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 divisor or a unit. In fact, we proved that $R$ is a $huz$-ring if and only 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 $R$.
%> http://ijmsi.ir/article-1-984-en.pdf
%P 1-8
%& 1
%!
%9 Research paper
%L A-10-2277-1
%+ Jordan university of science and technology
%G eng
%@ 1735-4463
%[ 2019