A generalization of exchange rings

In this paper, 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 [Formula: see text] arising in unit-regular rings, we explore configurations where such decompositions persist under weaker assumptions. By using outer inverse techniques, we characterize when such decompositions exist and connect these to partial unit-regularity and annihilator-stable (AS) elements. We further study inheritance properties of such weaker configuration conditions under extensions, corners, and matrix rings, and exhibit new examples of non-exchange rings, including certain group rings and pseudo-morphic constructions. Our results extend classical themes in the theory of regular, clean, and exchange rings and open new directions for studying structural approximations to unit-regularity.