We are doing a homework problem for our commutative algebra class, which asks us to prove:
Let $R$ be a commutative ring with $1$ containing finitely many minimal prime ideals $P_1, \dots, P_n$. If in addition $R$ satisfies that $R_M$ (the localization of $R$ at $M$) is a domain for all maximal ideals $M \subset R$, then $$ R \cong \frac{R}{P_1} \times \dots \times \frac{R}{P_n}. $$
If we could show that $R$ is Artinian, then we would be done by a result from class. But we think we don't have enough to show this.
Any help would be appreciated. Thanks.