Bull. Korean Math. Soc. 2007 Vol. 44, No. 4, 777-788 Published online December 1, 2007

Tai Keun Kwak Daejin University

Abstract : An endomorphism $\alpha$ of a ring $R$ is called {\it right {\rm (}left{\rm )} symmetric} if whenever $abc=0$ for $a, b, c \in R$, $ac\alpha(b)=0 \;(\alpha(b)ac=0)$. A ring $R$ is called {\it right {\rm (}left{\rm )} $\alpha$-symmetric} if there exists a right (left) symmetric endomorphism $\alpha$ of $R$. The notion of an $\alpha$-symmetric ring is a generalization of $\alpha$-rigid rings as well as an extension of symmetric rings. We study characterizations of $\alpha$-symmetric rings and their related properties including extensions. The relationship between $\alpha$-symmetric rings and (extended) Armendariz rings is also investigated, consequently several known results relating to $\alpha$-rigid and symmetric rings can be obtained as corollaries of our results.