Ring endomorphisms with nil-shifting property

Cohn called a ring $R$ is reversible if whenever $ab = 0,$ then $ba = 0$ for $a,bin R.$ The reversible property is an important role in noncommutative ring theory‎. ‎Recently‎, ‎Abdul-Jabbar et al‎. ‎studied the reversible ring property on nilpotent elements‎, ‎introducing‎ the 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 $ aalpha(b) = 0 $ (resp.‎, ‎$alpha(a)b = 0$) for nilpotents $a,b$ in $R$‎, ‎$ balpha(a) = 0 $ (resp.‎, ‎$ alpha(b)a= 0) $‎. ‎The characterization of $alpha$-nil-shifting rings and their related properties are investigated‎.