The X-semiprimeness of rings

For a nonempty subset [Formula: see text] of a ring [Formula: see text], the ring [Formula: see text] is called [Formula: see text]-semiprime if, given [Formula: see text], [Formula: see text] implies [Formula: see text]. 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 [Formula: see text] of a ring [Formula: see text], we offer a criterion for [Formula: see text] to be [Formula: see text]-semiprime. For a prime ring [Formula: see text], we characterizes Lie ideals [Formula: see text] of [Formula: see text] such that [Formula: see text] is [Formula: see text]-semiprime. Moreover, [Formula: see text]-semiprimeness of matrix rings, prime rings (with a nontrivial idempotent), semiprime rings, regular rings, and subdirect products are studied.