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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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When every localization of the polynomial ring over a ring has finitely many idempotents

Let $R$ be a commutative ring such that every localization ring $R_r$ has finitely many idempotents for each non nilpotent element $r\in R$. Why dose every localization ring $R[x]_{f(x)}$ have fini …