Generalizing a property of regular resp. finite semigroups a semigroup S is called E-(0-) inversive if for every a ∈S4(a ≠ 0) there exists x ∈Ssuch that ax (≠ 0) is an idempotent. Several characterizations are given allowing to identify the (completely, resp. eventually) regular semigroups in this class. The case that for every a ∈S4(≠ 0) there exist x,y ∈Ssuch that ax = ya(≠ 0) is an idempotent, is dealt with also. Ideal extensions of E- (0-)inversive semigroups are studied discribing in particular retract extensions of completely simple semigroups. The structure of E- (0-)inversive semigroups satisfying different cancellativity conditions is elucidated. 1991 AMS classification number: 20M10.