Examples of principal ideal local Artinian rings

Question

I want to find some examples of principal ideal local Artinian rings. That is, a commutative ring $R$ with identity with a unique maximal ideal, and this ideal is generated by a nilpotent element. (There is not any idempotent element in a local ring except $0$ and $1$, so under the Artinian condition, the maximal ideal cannot be generated by an idempotent one, and hence must be generated by a nilpotent one).

I have found three examples:

(1) Quotient ring $\mathbf{Z}/ p^d \mathbf{Z}$ with $d \ge 2$ and prime $p$;

(2) Quotient ring $K [x]/ \langle x^2 \rangle$ with a field $K$;

(3) Localization $(\mathbf{Z}/ p^d q \mathbf{Z})_q$ of the quotient ring $\mathbf{Z}/ p^d q \mathbf{Z}$ on $q$ with $d \ge 2$ and distinct primes $p$ and $q$.

But I realized that (3) is isomorphic to (1).

For some application, I need more examples (finite ring is more useful for me). But I cannot figure out any one else.

Many many thanks!

Answer 1

Well, most generally, if you take any principal maximal ideal $M$ of a commutative ring $R$, and select a nonzero integer $n > 1$, then $R/M^n$ is going to be an example.

Considering that all local principal ideal rings look like that (a ring whose ideals are precisely the powers of its maximal ideal) it's unlikely you'll get a much broader class of examples.

The trick, I suppose, is to be able to choose so that $M$ does interesting things. For example, if $M^2=M$ you only get a field. But if $R$ is a domain, then you're guaranteed that $M^{i+1}\subsetneq M^i$.

As it happens the DaRT query only yields two things, both of which you've covered already. But you could take any of the results of the PIDs that aren't fields query and do the construction I'm describing.