Akizuki–Hopkins–Levitzki theorem

Remarks

The condition that every prime ideal is maximal says that $R$ has Krull dimension $0$ when $R$ is not the zero ring. If $R$ is Artinian, each prime $P$ gives an Artinian domain $R/P$, which is a field, so $P$ is maximal. Then $R$ has finitely many maximal ideals $\mathfrak{m}_1, \dots, \mathfrak{m}_n$, and $(\mathfrak{m}_1 \cdots \mathfrak{m}_n)^k = 0$ for some $k$. The chain $R \supseteq \mathfrak{m}_1 \supseteq \mathfrak{m}_1 \mathfrak{m}_2 \supseteq \cdots \supseteq 0$ has factors that are vector spaces over fields $R/\mathfrak{m}_i$, and for vector spaces the two chain conditions agree, so $R$ has finite length and is Noetherian. For example, $\Z/12\Z$ is Artinian and Noetherian, while $\Z$ is Noetherian but has the non-maximal prime $(0)$, so it is not Artinian. For noncommutative rings, the Hopkins–Levitzki theorem says that every left Artinian ring is left Noetherian, while the converse needs extra hypotheses.

Going deeper

For a student

The theorem shows that Artinian rings are the Noetherian rings of dimension $0$, and it is often used together with the structure theorem for Artinian rings. A typical use: if $R$ is a Noetherian local ring and $I$ is an ideal whose radical is the maximal ideal, then $R/I$ is Artinian, so it has finite length. This fact is the start of dimension theory and of the Krull–Akizuki theorem.