Volume 14, Issue 2 (10-2019)                   IJMSI 2019, 14(2): 1-8 | Back to browse issues page

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Bataineh M, Abu-dawwas R. Graded r-Ideals. IJMSI. 2019; 14 (2) :1-8
URL: http://ijmsi.ir/article-1-984-en.html

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 $R$.

Type of Study: Research paper | Subject: Special

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