Volume 9 - Issue 2 (4) | PP: 80 - 92
Language : English

DOI : https://doi.org/10.31559/glm2020.9.2.4

DOI : https://doi.org/10.31559/glm2020.9.2.4

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#### On strong CNZ rings and their extensions

Received Date | Revised Date | Accepted Date | Publication Date |

6/1/2020 | 31/8/2020 | 7/9/2020 | 31/1/2021 |

##### 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.

**Keywords: **CNZ ring, reversible ring, matrix ring, polynomial ring

##### How To Cite This Article

Ahmed , C. A. K. (2021). On strong CNZ rings and their extensions . General Letters in Mathematics, 9 (2), 80-92, 10.31559/glm2020.9.2.4

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