I don't see how the axioms for a (general) unity ring preclude a zero divisor from having a one sided reciprocal as long as it does not have a zero cofactor on the opposite side of the reciprocal. However I can not find an example of such occurring in any ring. In other words can any one give me an example of a ring that has the elements a, b, c such that ab=0 and ac=1 or prove that this is not possible? Remember do not assume that the ring is commutative or finite.
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1An example of a right zero divisor that isn't a left zero divisor appears here – lulu Feb 13 '25 at 04:16
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Note that it is easy to give a one sided inverse for that example. – lulu Feb 13 '25 at 04:42