Let $K$ be a number field, $Z_K$ its ring of integers, and $p$ a rational prime number. Then $A_p = Z_K/(p)$ is a finite ${\mathbb F}_p$-algebra. Using ideal arithmetic in $Z_K$ and the Chinese remainder theorem it is easily checked that $A_p$ is the direct sum of finite fields if $p$ is unramified, and has additonal nil-rings as components if $p$ is ramified.

This decomposition result looks simple enough to have a direct and not too complicated proof. Basing it on the close relation between $A_p$ and ${\mathbb F}_p[X]/(f)$, where $f$ is the minimal polynomial of a generator of $K$, brings in problems with primes dividing the discriminant of $f$. Thus let me state my main question explicitly:

**Is there a simple proof that $A_p$ is a direct sum of finite fields and some easily described nil rings?**

In addition, I'd be grateful for pointers to the relevant literature, in particular to classification theorems of which the result above is a special case.