Ring endomorphisms with nil-shifting property

Subject Areas : Commutative algebra

C. A. K. Ahmed ¹ , R. T. M. Salim ²

1 - Department of Mathematics, University of Zakho, Kurdistan Region, Iraq
2 - Department of Mathematics, Faculty of Science, University of Zakho, Kurdistan Region, Iraq

Received: 2018-10-17 Accepted : 2019-05-26 Published : 2019-08-01

Keywords: CNZ ring&lrm, , &lrm, reversible ring&lrm, , &lrm, matrix ring&lrm, , &lrm, polynomial ring,

Abstract :

Cohn called a ring $R$ is reversible if whenever $ab = 0,$ then $ba = 0$ for $a,b\in R.$ The reversible property is an important role in noncommutative ring theory‎. ‎Recently‎, ‎Abdul-Jabbar et al‎. ‎studied the reversible ring property on nilpotent elements‎, ‎introducing‎the concept of commutativity of nilpotent elements at zero (simply‎, ‎a CNZ ring)‎. ‎In this paper‎, ‎we extend the CNZ property of a ring as follows‎: ‎Let $R$ be a ring and $\alpha$ an endomorphism of $R$‎, ‎we say that $ R $ is right (resp.‎, ‎left) $\alpha$-nil-shifting ring if whenever $ a\alpha(b) = 0 $ (resp.‎, ‎$\alpha(a)b = 0$) for nilpotents $a,b$ in $R$‎, ‎$ b\alpha(a) = 0 $ (resp.‎, ‎$ \alpha(b)a= 0) $‎. ‎The characterization of $\alpha$-nil-shifting rings and their related properties are investigated‎.

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