Abstract
K. I. Beidar and Y.-F. Lin have recently showed that under appropriate conditions a commutativity preserving map between (Jordan) algebras A and $$\mathcal{Q}$$ is of a standard form, unless it sends a certain subset of A, which one could describe (unless A is very special) as a “large” one, into the center of $$\mathcal{Q}$$ . We give a supplement to this statement by showing that this set often contains a nonzero ideal. In particular this makes it possible for us to give the definitive description of commutativity preservers in simple rings, as well as in prime rings provided that the map in question preserves commutativity in both directions.
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