Corollary 1

Every Artinian ring A is isomorphic to the direct composition of a finite family of Artinian local rings.

It follows from Proposition 9 and Proposition 8 (iii) and (iv) that, if $(\mathfrak{m}_i)_{1 \leq i \leq n}$ is the family of maximal ideals of $A$, the canonical mapping $A \to \prod_i A_{\mathfrak{m}_i}$ is bijective.