The X-semiprimeness of Rings

For a nonempty subset X of a ring R, the ring R is called X-semiprime if, given a∈ 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.