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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 finitely many idempotents for each non nilpotent polynomial $f(x)\in R[x]$?

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    $\begingroup$ By quotient ring $R_r$, do you mean $R/(r)$ or the localization? $\endgroup$
    – Asvin
    Commented Jul 4, 2020 at 17:32
  • $\begingroup$ @Asvin: thank you for your comment. It was corrected. $\endgroup$
    – Bazara
    Commented Jul 4, 2020 at 17:51

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