Weaker hypotheses than regularity suffice for left-right transfer of C4* properties in rings, with identified obstructions from square-free phenomena and permanence under matrix passage.
Nielsen, and Michał Ziembowski,Commuting idempotents, square-free modules, and the exchange property, Journal of Algebra444(2015), 52–80
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Left--right Transfer for $C4^{\ast}$-Rings Beyond the Regular Case
Weaker hypotheses than regularity suffice for left-right transfer of C4* properties in rings, with identified obstructions from square-free phenomena and permanence under matrix passage.