On Weak Symmetric Property of Rings

Abstract

A concept of a weak symmetric ring is defined by Ouyang and Chen, that is, a ring R is called weak symmetric if abc being nilpotent implies that acb is nilpotent for all a, b, c is an element of R. In this note we continue to study some extensions of weak symmetric rings and obtain some characterizations of weak symmetric rings from different perspectives. Among others it is proved that a ring R is weak symmetric if and only if for any nilpotent a is an element of R, aR is nil if and only if for any nilpotent a is an element of R, Ra is nil. It is showed that every 2-primal ring is weak symmetric. If the set of all nilpotent elements N(R) of R forms an ideal, then R and R/N(R) are weak symmetric. We also prove that R is a weak symmetric ring if and only if the ring M-(s)(R) of a special Morita context is weak symmetric if and only if the Dorroh extension D(R; Z) of R is weak symmetric.