P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a,

. 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

an endomorphism of R, we say that R is right (resp., left)

-shifting if whenever

(resp.,

) for a,

,

(resp.,

); and the ring R is called

-shifting if it is both left and right

-shifting. We investigate characterizations of

-shifting rings and their related properties, including the trivial extension, Jordan extension and Dorroh extension. In particular, it is shown that for an automorphism

of a ring R, R is right (resp., left)

-shifting if and only if Q(R) is right (resp., left)

-shifting, whenever there exists the classical right quotient ring Q(R) of R.