The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Proposition 10.59.6. Let $R$ be a ring. The following are equivalent:

  1. $R$ is Artinian,

  2. $R$ is Noetherian and $\dim (R) = 0$,

  3. $R$ has finite length as a module over itself,

  4. $R$ is a finite product of Artinian local rings,

  5. $R$ is Noetherian and $\mathop{\mathrm{Spec}}(R)$ is a finite discrete topological space,

  6. $R$ is a finite product of Noetherian local rings of dimension $0$,

  7. $R$ is a finite product of Noetherian local rings $R_ i$ with $d(R_ i) = 0$,

  8. $R$ is a finite product of Noetherian local rings $R_ i$ whose maximal ideals are nilpotent,

  9. $R$ is Noetherian, has finitely many maximal ideals and its Jacobson radical ideal is nilpotent, and

  10. $R$ is Noetherian and there are no strict inclusions among its primes.


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