Let $R$ a finite commutative unital ring.
Prove that an ideal of $R$ is irreducible if and only if it is prime.
It is clear that prime ideals are always irreducible.
But when irreducibility implies prime. It is known that it is true if $R$ is a PID. However, is it sufficient for $R$ to be a finite commutative unital ring?
Let $P$ an irreducible ideal such that $P=I_1\cap I_2$ for some $I_1,I_2$ ideals of $R$. Since $P$ is irreducible, $P=I_1$ or $P=I_2$.
If $P=I_1$, in particular, $I_1\subseteq P$. And if $P=I_2$, in particular, $I_2\subseteq P$. Therefore $P$ is prime.
But we have supposed that $P$ is the intersection of two ideals. Is this always possible/true in a finite commutative unital ring?