TITLE: Quotients of number rings
QUESTION [14 upvotes]: Hi,
Here's a question that comes up every now and then.  Of course, the quotient of a number ring (ring of integers of a number field) by an ideal $I$ is a finite (Artin) ring.  If we take $I$ to be the power of a prime, we obtain a finite local (Artinian) ring.  Is there a characterization of finite local rings which arise in this way?
In particular, can we obtain the quotient rings $k[x]/x^n$, where $k$ is a finite field?
Much thanks!

REPLY [4 votes]: This answer consists of a few remarks to put the "necessary condition" in context.  Recall:

Theorem: If $R$ is a Dedekind domain and $I$ is a nonzero ideal of $R$, then $R/I$ is a principal Artinian ring (i.e., an Artinian ring in which every ideal is principal). 

Using the factorization of ideals into primes, the Chinese Remainder Theorem, and the easy fact that a finite product of rings is principal Artinian iff each factor is principal Artinian, one reduces to the case of $I = \mathfrak{p}^a$ a prime power.  The ideals of $R/I$ correspond to the ideals of $R$ containing $\mathfrak{p}^a$, i.e., $R, \mathfrak{p},\ldots,\mathfrak{p}^a$.  So $R/I$ is certainly Artinian.  Moreover, $R/I = R_{\mathfrak{p}}/I_{\mathfrak{p}}$ is also a quotient of a DVR, hence of a PID, and any quotient of a principal ring is principal.  
In my experience it is traditional to emphasize not the previous result per se but the following consequence.

Theorem (C.-H. Sah) For an integral domain $R$, the following are equivalent:
  (i) $R$ is a Dedekind domain.
  (ii) Every ideal $I$ of $R$ can be generated by "$1+\epsilon$" elements: for all $0 \neq a \in I$, there exists $b \in I$ such that $I = \langle a,b \rangle$.

The implication (i) $\implies$ (ii) is immediate from the above.  The converse implication is also short but somewhat tricky, so I refer to $\S 20.5$ of my commutative algebra notes for the proof.
So it should be "well known" that the answer to the OP's question is not all finite rings.  Note also that the example $\mathbb{F}_q[x,y]/\langle x,y \rangle^2$ of a finite non-principal ring appears as an Exercise in $\S 16.3$ of my notes.  I am very sorry to say that my 264 pages of notes have no discussion of Zariski tangent spaces whatsoever: caveat emptor, to be sure.