  Regarding the question of how idempotent elements affect reversible property
of rings, we study a version of reversibility depending on idempotents. In this
perspective, we introduce {\it right} (resp., {\it left}) {\it $e$-reversible
rings}. We show that this concept is not left-right symmetric. Basic properties
of right $e$-reversibility in a ring are provided. Among others it is proved
that if $R$ is a semiprime ring, then $R$ is right $e$-reversible if and only
if it is right $e$-reduced if and only if it is $e$-symmetric if and only if it
is right $e$-semicommutative. Also, for a right $e$-reversible ring $R$, $R$ is
a prime ring if and only if it is a domain. It is shown that the class of right
$e$-reversible rings is strictly between that of $e$-symmetric rings and right
$e$-semicommutative rings.
