Structure theorem for Artinian rings

Remarks

In an Artinian ring the Jacobson radical $\mathfrak{m}_1 \cap \cdots \cap \mathfrak{m}_n$ is nilpotent, so $\mathfrak{m}_1^k \cdots \mathfrak{m}_n^k = 0$ for some $k$. The ideals $\mathfrak{m}_i^k$ are pairwise coprime, so the Chinese remainder theorem gives $R \cong \prod_i R/\mathfrak{m}_i^k$. One then checks that $R/\mathfrak{m}_i^k \cong R_{\mathfrak{m}_i}$. For example, $\Z/12\Z \cong \Z/4\Z \times \Z/3\Z$, and these factors are the localizations at the maximal ideals $(2)$ and $(3)$. The decomposition is unique: in any decomposition of $R$ as a finite product of rings of local rings, the factors are, up to isomorphism and order, the rings $R_{\mathfrak{m}_i}$.

Going deeper

For a student

Together with the Akizuki–Hopkins–Levitzki theorem, this reduces many questions about zero-dimensional Noetherian rings to the local case. Geometrically, the spectrum of an Artinian ring is a finite discrete set of points, and the ring is the product of the "fat points" at each of them. An Artinian local ring is the kind of ring met as $R/I$ for a Noetherian local ring $R$ and an ideal $I$ primary to the maximal ideal.