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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