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!