Iranian Journal of Mathematical Sciences and Informatics
مجله علوم ریاضی و انفورماتیک
IJMSI
Basic Sciences
http://ijmsi.ir
1
admin
1735-4463
2008-9473
8
10.29252/ijmsi
14
8888
13
en
jalali
1398
7
1
gregorian
2019
10
1
14
2
online
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fulltext
en
Graded r-Ideals
تخصصي
Special
پژوهشي
Research paper
<p>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<br>
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 $aP=aRbigcap P$<br>
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,<br>
$P$ 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 $R$.</p>
Graded prime ideals, Graded r-ideals.
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http://ijmsi.ir/browse.php?a_code=A-10-2277-1&slc_lang=en&sid=1
R.
Abu-dawwas
rrashid@yu.edu.jo
`10031947532846007789`

10031947532846007789
No
Department of Mathematics, Yarmouk University, Jordan.
M.
Bataineh
msbataineh@just.edu.jo
`10031947532846007790`

10031947532846007790
Yes
Department of Mathematics and Statistics, Jordan University of Science and Technology, Jordan.