This post will assume familiarity with some basic notions from commutative algebra such as localization for modules and exact sequences. Here we will mean “ring” to mean a commutative ring with unity.
This post is about to the “Chinese Remainder Theorem.) however, I prefer a renaming— while Sunzi’s identity is unknown, I do not think “Sunzi’s remainder theorem” will cause any great confusion.
Anyhow, the theorem is best remembered by the informal statement “you can simultaneously solve a system of modular conditions, given that the moduli are coprime.” We’ll consider a mild generalization to arbitrary rings, granted that we can make sense of “coprime” outside of principal ideal domains, leading us to the following definition:
Definition 1 Two ideals of a ring are said to be coprime if .
One can easily verify that this definition is “correct” since it is both well defined, and also agrees with our usual notion of coprime numbers, since the condition is equivalent to the existence of and so that .
Now, we can state the theorem:
Main Theorem the map given by
is surjective when all are pairwise coprime. (It is also the case that
but this is much less fun to prove.)
If we take , then this essentially says that the map is surjective when are coprime. This ensures some solution to the set of modular conditions.
There is a standard proof for the main theorem, discoverable by any cursory google search. It is essentially the same as the special case for , using Bezout’s identity. However, I consistently forget this proof since I did not think of it myself, and no part of my being wishes to be reminded of this fact. Consequently, I want to present a more interesting one using a hint from Eisenbud’s “Commutative Algebra With a View Towards Algebraic Geometry.” We will need the notion of localization for modules, and some of its basic properties. In particular, we will need to use the fact that localization is “exact,” and that surjectivity is a “local property,” statements that will be made precise immediately:
given any -module homomorphism , localization at induces a new -module homomorphism , defined in the most natural way one can imagine: . Presently, we only care about localization at prime ideals, so in the case where for a prime ideal , we will denote the map by and the localization of by .
If I were a better person I would have made clear how the localization construction works with modules, but one must have faith that all is good in the world. A nice reference would be Atiyah-Macdonald’s classical text, from which we will borrow the statement and proof of the following proposition:
Proposition 2: if
is exact at , then
is exact as well.
proof: The first inclusion is immediate since . For the other direction, let . Then there is some so that , so by our hypothesis. Now, we simply take so that . But then which was our definition for , proving the proposition.
With all of this sorted out we can finally prove that surjectivity is a local property, meaning that if it holds for localizations at every prime ideal, it holds for the entire ring! We will need one corollary from the previous proposition: quotients “commute” with taking localizations, or rather , which can be shown by considering the sequence .
Here is the final lemma needed before our proof of the main theorem:
Lemma 3: As before, consider a sequence of modules
We claim that if
is exact for every maximal ideal of , then the first sequence is exact as well.
Proof: Again the first inclusion is clear. This means that is well defined, and is at the localization, since by proposition 2 (applied twice) and the hypothesis of local exactness.
Thus, it will suffice to show that if for all maximal ideals, then . This this will show that .
I will not lie, this is an annoying verification, so the following is hasty: assume that exists so its annihilator, , is contained in some maximal ideal since . If we localize at this maximal ideal, will be zero by hypothesis. But then gets eaten by something outside of , a contradiction.
Sigh! the deed is done, we have all that we require to show that surjectivity is local, since one need only remark that if the sequence
is exact at every maximal ideal, so is the “non-localized version.”
We proceed now to the allegedly interesting proof of the main theorem:
Main Theorem: the map given by
is surjective whenever all are pairwise coprime.
proof: localize at an arbitrary maximal ideal . We now note that if none among the belong to , the map is trivially surjective. On the other hand suppose that , meaning that no other among the family can also belong to since this would contradict coprimality. Then, . But then is really just the natural surjection . Since was arbitrary we have by lemma , that is a surjective map.