Communications of the
Korean Mathematical Society
CKMS

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

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Commun. Korean Math. Soc. 2010; 25(3): 349-364

Printed September 1, 2010

https://doi.org/10.4134/CKMS.2010.25.3.349

Copyright © The Korean Mathematical Society.

Ring endomorphisms with the reversible condition

Muhittin Baser, Fatma Kaynarca, and Tai Keun Kwak

Kocatepe University, Kocatepe University, Daejin University

Abstract

P. M. Cohn called a ring $R$ $reversible$ if whenever $ab=0$, then $ba=0$ for $a, b \in R$. Commutative rings and reduced rings are reversible. In this paper, we extend the reversible condition of a ring as follows: Let $R$ be a ring and $\alpha$ an endomorphism of $R$, we say that $R$ is $right$ (resp., $left$} $\alpha$-$shifting$ if whenever $a\alpha(b)=0$ (resp., $\alpha(a)b=0$) for $a, b \in R$, $b\alpha(a)=0$ (resp., $\alpha(b)a=0$); and the ring $R$ is called $\alpha$-$shifting$ if it is both left and right $\alpha$-shifting. We investigate characterizations of $\alpha$-shifting rings and their related properties, including the trivial extension, Jordan extension and Dorroh extension. In particular, it is shown that for an automorphism $\alpha$ of a ring $R$, $R$ is right (resp., left) $\alpha$-shifting if and only if $Q(R)$ is right (resp., left) $\bar{\alpha}$-shifting, whenever there exists the classical right quotient ring $Q(R)$ of $R$.

Keywords: ring endomorphism, reduced ring, reversible ring, trivial extension, classical right quotient ring

MSC numbers: Primary 16W20, 16U80; Secondary 16S36