Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is Noetherian). If this is not true in general, what if I also assumed $S$ is commutative?
2026-03-27 05:38:11.1774589891
Are module finite algebras over Noetherian semiperfect rings again semiperfect?
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