Structure of Finite Commutative Rings

Solution 1:

Sure. A ring homomorphism $\prod_i R_i \to S$ corresponds to a decomposition $S =\prod_i S_i$ and a family of ring homomorphisms $R_i \to S_i$ (namely $S_j=S \overline{e_j}$ where $\overline{e_j}$ is the image of the canonical idempotent $e_j \in \prod_i R_i$). Thus, an algebra over a product is a product of algebras.

A finite ring has some characteristic $n \neq 0$, i.e. is an algebra over $\mathbb{Z}/n \cong \prod_p \mathbb{Z}/p^{v_p(n)}$.

Solution 2:

A finite commutative ring is artinian, so it is a finite product of finite artinian local rings. Let $R$ be a finite commutative local ring and $\phi:\mathbb Z\to R$ the canonical ring homomorphism ($n\mapsto n1_R$). Then $\ker \phi=n\mathbb Z$, $n>1$. It follows that $\mathbb Z/n\mathbb Z$ is isomorphic to a subring of $R$. Since the idempotents of $R$ are trivial, the same holds for $\mathbb Z/n\mathbb Z$. This implies that $n$ is a power of a prime, and we are done. (More on finite local rings you can find here.)