Let $R$ be a commutative unital ring. Let's call a non-zero non-unit $a \in R$
irreducible if $a=bc$ implies that (either) $b$ or $c$ is a unit,
prime if $a \mid bc$ implies that $a \mid b$ or $a \mid c$.
Question: Is there an example for a finite $R$ that contains an element that is irreducible but not prime? So far I know that $R$ can't be a principal ideal ring because it would imply "irreducible $\implies$ prime". Is this ring, that is $\mathbb{F}_2[x,y]/(x^2,xy,y^2)=\mathbb{F}_2[\tilde{x},\tilde{y}]=\{0,\,1,\,\tilde{x},\,\tilde{y},\,1+\tilde{x},\,1+\tilde{y},\,\tilde{x}+\tilde{y},\,1+\tilde{x}+\tilde{y}\}$, a suitable example? It seems that $\tilde x$ is irreducible: Any factorization of $\tilde x$ must be of the form $\tilde x = \tilde x (1+a\tilde x+b\tilde y)$ with $a,b\in \mathbb F_2$, and $(1+a\tilde x+b\tilde y)$ is self-inverse. On the other hand, $\tilde x$ is not prime because $\tilde x \mid \tilde y^2 (= 0)$, but $\tilde x \nmid \tilde y$. More precisely, all the multiples of $\tilde x$ are $0$ and $\tilde x$ itself.