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, ‎abduljabbar 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) $alpha$nilshifting 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 $alpha$nilshifting rings and their related properties are investigated‎.