- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnline Submission ㆍMy Manuscript - For Reviewers - For Editors
 On radically-symmetric ideals Commun. Korean Math. Soc. 2011 Vol. 26, No. 3, 339-348 https://doi.org/10.4134/CKMS.2011.26.3.339Published online September 1, 2011 Ebrahim Hashemi Shahrood University of Technology Abstract : A ring $R$ is called symmetric, if $abc=0$ implies $acb=0$ for $a,b,c\in R$. An ideal $I$ of a ring $R$ is called symmetric (resp. radically-symmetric) if $R/I$ (resp. $R/\sqrt{I}$) is a symmetric ring. We first show that symmetric ideals and ideals which have the insertion of factors property are radically-symmetric. We next show that if $R$ is a semicommutative ring, then $T_n(R)$ and $R[x]/(x^n)$ are radically-symmetric, where $(x^n)$ is the ideal of $R[x]$ generated by $x^n$. Also we give some examples of radically-symmetric ideals which are not symmetric. Connections between symmetric ideals of $R$ and related ideals of some ring extensions are also shown. In particular we show that if $R$ is a symmetric (or semicommutative) $(\alpha,\delta)$-compatible ring, then $R[x;\alpha,\delta]$ is a radically-symmetric ring. As a corollary we obtain a generalization of [13]. Keywords : insertion of factors property, $(\alpha,\delta)$-compatible ideals, $\alpha$-rigid ideals, Ore extensions, symmetric rings, semicommutative rings MSC numbers : 16S36, 20M11, 20M12 Downloads: Full-text PDF