Why does the intersection of these ideals equal their product?

Question

I was reading a proof of the above theorem (1.6.7 Theorem) from here, but there was something that confused me. The proof says $R$ has finitely many maximal ideals $M_1, \ldots ,M_r$, and the intersection of these ideals is nilpotent. It goes on to say "By the Chinese remainder theorem, the intersection of the $M_i$ coincides with their product." I'm not quite sure how the Chinese remainder implies that... so I was wondering if anybody could clarify that to me.

Answer 1

Any two maximal ideals $M_1$, $M_2$ are comaximal, i.e. $M_1+M_2=R$. This implies that $m_1 + m_2 = 1$ for some $m_1 \in M_1$, $m_2 \in M_2$. If $m \in M_1 \cap M_2$, then $mm_1+mm_2 = m \in M_1M_2$. So $M_1\cap M_2 \subseteq M_1M_2$, and the other inclusion is always true. This generalizes to any number of maximal ideals.

Answer 2

This is not the Chinese Remainder Theorem. It is the elementary and easy-to-prove fact (sometimes included into the statement of CRT) that for pairwise coprime ideals $I_1,\dotsc,I_n$ we have $I_1 \cap \dotsc \cap I_n = I_1 \cdot \dotsc \cdot I_n$.