P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a, $b;{in};R$. Commutative rings and reduced rings are reversible. In this paper, we extend the reversible condition of a ring as follows: Let R be a ring and $alpha$ an endomorphism of R, we say that R is right (resp., left) $alpha$-shifting if whenever $a{alpha}(b);=;0$ (resp., $alpha{a)b;=;0$) for a, $b;{in};R$, $b{alpha}{a);=;0$ (resp., $alpha(b)a;=;0$); and the ring R is called $alpha$-shifting if it is both left and right $alpha$-shifting. We investigate characterizations of $alpha$-shifting rings and their related properties, including the trivial extension, Jordan extension and Dorroh extension. In particular, it is shown that for an automorphism $alpha$ of a ring R, R is right (resp., left) $alpha$-shifting if and only if Q(R) is right (resp., left) $ar{alpha}$-shifting, whenever there exists the classical right quotient ring Q(R) of R.