Symmetric property of rings with respect to the Jacobson radical

Tugce Pekacar Calci; Sait Halicioglu; Abdullah Harmanci

doi:10.4134/CKMS.c170473

Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2019; 34(1): 43-54

Online first article June 14, 2018 Printed January 31, 2019

https://doi.org/10.4134/CKMS.c170473

Symmetric property of rings with respect to the Jacobson radical

Tugce Pekacar Calci, Sait Halicioglu, Abdullah Harmanci

Ankara University; Ankara University; Hacettepe University

Abstract

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Abstract

Let $R$ be a ring with identity and $J(R)$ denote the Jacobsonradical of $R$, i.e., the intersection of all maximal left ideals of $R$. A ring $R$ is called {\it $J$-symmetric} if for any $a,b,c\in R$, $abc = 0$ implies $bac \in J(R)$. We prove that some results of symmetric rings can be extended to the $J$-symmetric rings for this general setting. We give many characterizations of such rings. We show that the class of $J$-symmetric rings lies strictly between the class of symmetric rings and the class of directly finite rings.

Keywords: symmetric ring, $J$-symmetric ring, ring extension

MSC numbers: 13C99, 16D80, 16U80