Let $R$ be a Dedekind domain and its Jacobson radical is non-zero $\mathrm{Jac}(R)\neq 0$. Why $\dim(R/\mathrm{Jac}(R))=0$?
2026-03-26 21:26:09.1774560369
Why $\dim(R/\mathrm{Jac}(R))=0$ for Dedekind domain $R$ with $\mathrm{Jac}(R)\neq 0$?
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Since Dedekind domains are one dimensional, every non zero prime ideal is maximal. If $Jac(R)\not=0$, then every prime ideal of $ R/Jac (R) $ must be maximal. Thus, $R/Jac (R) $ is zero dimensional.