Let $R$ be a commutative ring with unity. $R$ is regular if for each $a\in R$, there exists $b\in R$ such that $a=aba$. It is local it has a unique maximal ideal.
(James L. Fisher , Finite Principal Ideal Ring, Canad. Math. Bull. Vol. 19 (3), 1976) proves the following:
THEOREM 2. Let $R$ be a finite principal ideal ring. Then $R$ is an ideal direct sum $R_1\oplus\cdots \oplus R_k\oplus N$ where $R_i, i = l,\ldots, k$ are primary principal ideal rings and $N$ is a nilpotent principal ideal ring.
I am trying to check for examples of commutative rings satisfying this results but most of them are regular ring.
My question: is there an example of a finite non-regular ring which is direct product of local rings one of which is a field and the other is not a field?