TITLE: Structure of Finite Commutative Rings
QUESTION [21 upvotes]: Is every finite commutative ring $A$ a direct product of finite algebras over $\mathbb Z/p^n$?

REPLY [13 votes]: 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.)