# Generalization of the Structure theorem for artinian rings?

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 $\prod_{i=1}^nA_{(p_i)}$?

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Exactly on how many examples did you try this on? :) –  Mariano Suárez-Alvarez Nov 21 '10 at 4:03
ti4: Did you ask this question math.stackexchange.com/questions/10980/… ? –  Hailong Dao Nov 21 '10 at 4:03
Did you mean to assume that $A$ was Artinian? –  Karl Schwede Nov 21 '10 at 4:28
Mariano: Perhaps I should've added that I thought it was wrong but could'nt find a convincing counterexample :) –  Pandamic Nov 21 '10 at 11:31
Hailong: Yes I did, it is almost the same question, only this is a little weaker perhaps. –  Pandamic Nov 21 '10 at 11:33

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).
You probably need to think carefully about what it means to localize at the prime $(0)$. –  Todd Trimble Nov 21 '10 at 11:42