I’m going to start off weird, with no motivation (because the reader will be able to recognize the statement almost immediately):
Theorem: Suppose are relatively prime positive integers . Then the system , has a unique solution mod .
That’s the Chinese remainder theorem. In this brief post, I’ll write down other ways of writing down the same statement, for comparison (and actually simply to have a collection of the statements). I refer the reader to Milne’s algebraic geometry notes for proofs, which I won’t be providing here (since that’s not the main purpose).
First Method: Let . Then is isomorphic as a ring to .
This is really just a homework exercise in basic algebra. We are going to generalize this. All rings are going to be commutative and unital.
Definition: An ideal of a ring is a subset of such that:
- is a subgroup of as a group under addition.
- If and , then .
Define to be . This is an ideal (exercise). Two ideals are coprime if is isomorphic to . Then (from Milne’s book):
Second Method: Suppose is a commutative unital ring. Then for ideals such that if , then , the map
is surjective, and the kernel is .
One more method is using scheme theory. An ideal is prime if is an integral domain. Let be a ring. denotes the collection of all prime ideals of . This can be equipped with a topology, called the Zariski topology. The basis of open sets for this topology are of the following form, for :
We may define a presheaf of rings on via . Then the Chinese remainder theorem follows from the following statement:
Third Method: The presheaf is actually a sheaf. (See here for a proof.)
The Chinese remainder theorem follows from applying the gluing property to disjoint open subsets. I’d love to know if there are more statements equivalent to the Chinese remainder theorem – these are just some that I can recall off the top of my head!