P. M. Cohn called a ring R

if whenever

= 0, then

= 0 for

. In this paper, we study an extension of a reversible ring with its endomorphism. An endomorphism

of a ring R is called

(resp.,

)

if whenever

= 0 (resp.,

= 0) for

,

= 0. A ring R is called

(resp.,

)

if there exists a strong right (resp., left) reversible endomorphism

of R, and the ring R is called

if R is both strong left and right

-reversible. We investigate characterizations of strong

-reversible rings and their related properties including extensions. In particular, we show that every semiprime and strong

-reversible ring is

-rigid and that for an

-skew Armendariz ring R, the ring R is reversible and strong

-reversible if and only if the skew polynomial ring R[

] of R is reversible.