Left--right Transfer for $C4^{\ast}$-Rings Beyond the Regular Case

Abstract

The first structural fact is that regularity is sufficient for left--right symmetry of the strongly \(C4^{\ast}\) condition. It is not necessary for the definition itself and is too strong for classification. The problem is therefore to determine which weaker hypotheses still force right \(C4^{\ast}\)-type conditions to pass to the left side, and which obstructions prevent such transfer. We study right \(C4^{\ast}\)-rings, strongly right \(C4^{\ast}\)-rings, and right semi-weak-CS \(C4^{\ast}\)-rings under hypotheses strictly weaker than regularity. The method does not repeat the regular decomposition argument. Instead, it isolates a transfer mechanism based on orthogonal decomposition, corner control, and summand-square-free separation. This yields sufficient conditions for left--right transfer beyond the regular case and also identifies necessary obstruction patterns. In particular, we determine settings in which one-sided \(C4^{\ast}\)-behavior forces two-sided \(C4^{\ast}\)-behavior, and settings in which the implication fails. A second aim is exact separation. We show that failure of transfer is structural, not accidental: it is encoded by persistent one-sided square-free phenomena and by the breakdown of compatible corner decompositions. This produces counterexample criteria rather than isolated examples. A third aim is permanence. The transfer principles are formulated so as to persist under matrix and full-corner passage whenever the relevant hypotheses are stable. Thus the symmetry problem is placed in a Morita-type framework, although full Morita invariance is not asserted for every \(C4^{\ast}\)-condition. The paper therefore divides the symmetry problem into transfer, obstruction, and permanence, and thereby sharpens the regular theory, delimits its range, and gives a classification scheme for the left--right behavior of \(C4^{\ast}\)-type rings.