Let $A$ be a commutative ring with identity. If $A$ has finite number of prime ideals $p_1,…p_n$ and moreover $\prod_{i=1}^n p_i^{k_i} = 0$ for some $k_i$. Are the prime ideals necessarily maximal?
[Math] Ring with finitely many prime ideals with an extra condition. Are they maximal
commutative-algebraring-theory
Best Answer
No, but the counterexample is trivial. Take any integral domain with finitely many prime ideals which is not a field. For example, the localization $\mathbb{Z}_{(p)}$ of the integers at a prime p. The zero ideal is non-maximal and prime so, trivially, $\prod_{i=1}^np_i=0$. Maybe this isn't exactly what you were meaning to ask?