Refaad for Studies, Research and Developement
On strong CNZ rings and their extensions
2519-9269 (Print)
2519-9277 (Online)
Volume 9
Issue 2
2020
Dec
0
On strong CNZ rings and their extensions
80
92
English
Chenar Abdul Kareem AhmedDepartment of Mathematics, Faculty of Science, University of Zakho, Kurdistan Region, Iraq chenar.ahmed@uoz.edu.krd
https://doi.org/10.31559/glm2020.9.2.4
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.
CNZ ring; reversible ring; matrix ring; polynomial ring
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