Bulletin of the
Korean Mathematical Society BKMS

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.

Keywords: reduced rings, symmetric rings, (extended) Armendariz rings

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