  For a nonempty subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime
if, given $a\in R$, $aXa=0$ implies $a=0$. This provides a proper class of
semiprime rings. First, we clarify the relationship between idempotent
semiprime and unit-semiprime rings. Secondly, given a Lie ideal $L$ of a ring
$R$, we offer a criterion for $R$ to be $L$-semiprime. For a prime ring $R$, we
characterizes Lie ideals $L$ of $R$ such that $R$ is $L$-semiprime. Moreover,
$X$-semiprimeness of matrix rings, prime rings (with a nontrivial idempotent),
semiprime rings, regular rings, and subdirect products are studied.
