Let $A$ be a commutative ring with identity. If $A$ is a ring with only a finite set of prime ideals $p\_1...p\_n$ and moreover $\prod\_{i=1}^n p\_i^{k\_i}=0$ for some k\_i. Is $A$ then isomorphic to $\prod\_{i=1}^nA\_{(p\_i)}$?



Generalization of the Structure theorem for artinian rings?
No. Let $A$ be a DVR. It has two prime ideals: the maximal ideal $p\_1=\mathfrak m\subset A$ and $p\_2=(0)\subset A$. So, $p\_1p\_2=0$, but $A$ is not a product (of two local rings).


