# Examples of principal ideal local Artinian rings

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!

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.
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 $$M^{i+1}\subsetneq M^i$$.