Abstract
Several examples are constructed, including a finite ring which cannot be embedded in matrices over any commutative ring, a semiprime PI ring with no classical ring of quotients, an example showing that the property of having all regular elements invertible is not inherited by matrix ringsMn(R), and a prime PI ringR with an idempotente such thatR/ReR has finitely generated projective modules not induced by any finitely-generated projective R-module.
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