Abstract:

The notion of weakly symmetric α-rings, which is rings with weakly symmetric endomorphismα, is introduced, and the study of weakly symmetric rings is extended. The relationships between weakly symmetric α-rings and related rings are discussed and some extensions of weakly symmetric α-rings are investigated. It is showed that (1) Let α be an endomorphism of a ring R. Then R is α-rigid if and only if R is weakly symmetric α-ring and aRα(a)∈nil(R) implies a=0 for any a∈R; (2) Let R be a semicommutative ring and α an endomorphism of R. Then R is weakly symmetric α-ring if and only if R［x］ is weakly symmetric -ring.

Key words: symmetric ring, weakly symmetric ring, weakly symmetric α-ring, α-rigid ring