§ Proof of chinese remainder theorem on rings
§ General operations on ideals
We have at our hands a commutative ring , and we wish to study the ideal
structure on the ring. In particular, we can combine ideals in the following
We have the containment:
§ is a ideal,
it's not immediate from the definition that is an ideal. The idea is
that given a sum , we can write each ,
since the ideal is closed under multiplication with . This gives
us . Similarly, we can interpret .
Hence, we get the containment .
Immediate from the inclusion function.
Immediate from inclusion
§ CRT from an exact sequence
There exists an exact sequence:
We are forced into this formula by considerations of dimension. We know:
By analogy to euler characteristic which arises from homology, we need to have
in the middle of our exact sequence. So we must have:
Now we need to decide on the relative ordering between and .
Thus, the exact sequence must have in the image of . This
forces us to arrive at:
- There is no universal way to send . It's an unnatural operation to restrict the direct sum into the intersection.
- There is a universal way to send : sum the two components. This can be seen as currying the addition operation.
The product ideal plays no role, since it's not possible to define a
product of modules in general (just as it is not possible to define
a product of vector spaces). Thus, the exact sequence better involve
module related operations. We can now recover CRT: