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!

Cheers,

S.D.

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I love this! Everything is so creative and entertaining to read! Keep up the good work! 🙂

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