On strong CNZ rings and their extensions

Abstract EN

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 2 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 of a ring R is called strong right ( resp., left) CNZ if whenever a (b) = 0(resp., (a)b = 0 ) for a, b 2 N(R) ba = 0.

A ring R is called strong right (resp., left) -CNZ if there exists a strong right (resp., left) CNZ endomorphism of R, and the ring R is called strong - CNZ if R is both strong left and right - CNZ.

Characterization of strong - 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 - CNZ rings are studied.