Shortest proof of the Chinese remainder theorem

Question

I was wondering about what is the shortest proof of the Chinese remainder theorem, so I can remember it for a test, and save my time. Can you help me with that?

Answer 1

I think the proof using localization is quite short, i.e. if $I_1,\dots,I_n$ are pairwise comaximal then any maximal ideal contains at most one of these ideals, hence the map $R\to R/I_1\times\dots\times R/I_n$ is surjective at the maximal ideals and therefore surjective itself.
In the case of $R=\Bbb Z$ there is also a short proof using the following argument: If $a,b$ are coprime then the map $\Bbb Z/ab\to \Bbb Z/a\times \Bbb Z/b$ is injective. As the two sets have the same number of elements it is also surjective (it also works for more that two elements).