Abstract
Lambek called a ring R symmetric if whenever abc = 0, then acb = 0 for a, b, c is an element of R. In this paper, we study an extension of symmetric ring with its endomorphism. An endomorphism alpha of a ring R is called strong right (resp., left) symmetric if whenever ac alpha(b) = 0 (resp., alpha(b)ac = 0) for a, b, c is an element of R, abc = 0. A ring R is called strong right (resp., left) alpha-symmetric if there exists a strong right (resp., left) symmetric endomorphism alpha of R, and the ring R is called strong alpha-symmetric if R is both strong left and right alpha-symmetric. We study characterizations of strong alpha-symmetric rings and their related properties including extensions. In particular, we show that every semiprime and strong alpha-symmetric ring is alpha-rigid and we prove that if R is an alpha-skew Armendariz ring, then R is symmetric and strong alpha-symmetric if and only if the skew polynomial ring R[x; alpha] of R is symmetric.