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 7 14 8888 13 en jalali 1398 7 1 gregorian 2019 10 1 14 2 online 1 fulltext
en Graded r-Ideals تخصصي Special پژوهشي Research paper <p>Let \$G\$ be a group with identity \$e\$ and \$R\$ be a commutative<br> \$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<br> graded \$r\$-ideal if whenever \$a, bin h(R)\$ such that \$abin P\$ and \$Ann(a)={0}\$,<br> then \$bin P\$. We study and investigate the behavior of graded \$r\$-ideals to introduce<br> &nbsp;several results. We introduced several characterizations for graded \$r\$-ideals;<br> &nbsp;we proved that \$P\$ is a graded \$r\$-ideal of \$R\$ if and only if \$aP=aRbigcap P\$<br> &nbsp;for all \$ain h(R)\$ with \$Ann(a)={0}\$. Also, \$P\$ is a graded \$r\$-ideal of \$R\$<br> &nbsp;if and only if \$P=(P:a)\$ for all \$ain h(R)\$ with \$Ann(a)={0}\$. Moreover,<br> &nbsp;\$P\$ is a graded \$r\$-ideal of \$R\$ if and only if whenever \$A, B\$ are graded ideals of<br> &nbsp; \$R\$ such that \$ABsubseteq P\$ and \$Abigcap r(h(R))neqphi\$, then \$Bsubseteq P\$.<br> &nbsp; &nbsp;In this article, we introduce the concept of \$huz\$-rings. A graded ring \$R\$<br> &nbsp; &nbsp;is said to be \$huz\$-ring if every homogeneous element of \$R\$ is either a zero<br> &nbsp; &nbsp; divisor or a unit. In fact, we proved that \$R\$ is a \$huz\$-ring if and only<br> &nbsp; &nbsp; &nbsp;if every graded ideal of \$R\$ is a graded \$r\$-ideal. Moreover, assuming that<br> &nbsp; &nbsp; &nbsp;\$R\$ is a graded domain, we proved that \${0}\$ is the only graded \$r\$-ideal of \$R\$.</p> Graded prime ideals, Graded r-ideals. 1 8 http://ijmsi.ir/browse.php?a_code=A-10-2277-1&slc_lang=en&sid=1 M. Bataineh msbataineh@just.edu.jo `10031947532846006431` 10031947532846006431 Yes jordan university of science and technology R. Abu-dawwas rrashid@yu.edu.jo `10031947532846006432` 10031947532846006432 No yarmouk university