On strong CNZ rings and their extensions.

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 ∝ of a ring R is called strong right ( resp., left) CNZ if whenever a∝(b) = 0(resp., ∝(a)b = 0 ) for a, b ∈ 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 U 2 (R) is a CNZ ring. Furthermore extensions of strong ∝- CNZ rings are studied. [ABSTRACT FROM AUTHOR]

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