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Let $k$ be some field, and let $v,v',v''$ be three distinct points in $k^3$. Let $\mathfrak{m}_v = (X_1 - v_1,X_2 - v_2,X_3 - v_3)$ be the ideal in $k[X_1,X_2,X_3]$ corresponding to the polynomials vanishing at $v$. We know that $\mathfrak{m}_v$ is a maximal ideal. I want to show that $\mathfrak{m}_v \cdot \mathfrak{m}_{v'} \cdot \mathfrak{m}_{v''} = \mathfrak{m}_v \cap \mathfrak{m}_{v'} \cap \mathfrak{m}_{v''}$. The inclusion "$\subset$" is always trivially true.

I'm not too sure how to approach the other direction. I know that for only two distinct maximal ideals $I,J$, they are necessarily comaximal and we do have $I \cdot J = I \cap J$. However, I'm not sure this generalizes to three ideals, since the product ideals $(I\cdot J)\cdot K$ and $I\cdot J\cdot K$ are seemingly distinct.

Any help would be gladly appreciated!

Alex Provost
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2 Answers2

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Given a finite collection of pairwise comaximal ideals, their product always equals their intersection. This is Proposition 1.10 in Atiyah-Macdonald Introduction to Commutative Algebra. The proof is by induction on the number of ideals.

Brenin
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Since the OP might not have access to Atiyah-MacDonald, I'll give a hint: It is induction, but it takes a moment's thought to get it to work. As atricolf mentioned, this holds for any finite number of coprime ideals. Say $\mathfrak{a}_1,\ldots,\mathfrak{a}_n$ are coprime and the result holds for $n-1$ ideals ($n>2$). Then $\bigcap_{i=2}^n\mathfrak{a_i} = \prod_{i=2}^n \mathfrak{a_i}$. Now, show $\mathfrak{a}_1 + \prod_{i=2}^n \mathfrak{a}_i = (1)$ and use the induction hypothesis again.

John Myers
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