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. پرونده مقاله