A Generalization of Exchange Rings

We investigate structural decompositions in rings that reflect partial forms of unit-regularity, focusing on containment conditions between principal and idempotent-generated ideals. Motivated by canonical decompositions such as R = Ra ⨁ R(a − u) arising in unit-regular rings, we explore configurations where such decompositions persist under weaker assumptions. Using outer inverse techniques, we characterize when such decompositions exist and connect these to partial unit-regularity and annihila