Abstract

T.K. Kwak and Y. Lee called a ring R satisfy the commutativity of nilpotent elements at zero[1] if ab = 0 for a, b ∈ N(R) implies ba = 0. For simplicity, a ring R is called CNZ if it satisfies the commutativity of nilpotent elements at zero. In this paper we study an extension of a CNZ ring with its endomorphism. An endomorphism α of a ring R is called strong right ( resp., left) CNZ if whenever aα(b) = 0(resp., α(a)b = 0 ) for a, b ∈ N(R) ba = 0. A ring R is called strong right (resp., left) α-CNZ if there exists a strong right (resp., left) CNZ endomorphism α of R, and the ring R is called strong α- CNZ if R is both strong left and right α- CNZ. Characterization of strong α- CNZ rings and their related properties including extensions are investigated . In particular, it’s shown that a ring R is reduced if and only if U2(R) is a CNZ ring. Furthermore extensions of strong α- CNZ rings are studied.