We will mostly adopt the Bourbaki definitions for the preliminary material. An ideal in a ring is prime if is an integral domain. Two ideals are relatively prime if A multiplicative subset of a set is a submagma of In a ring, we assume it is a submagma with respect to multiplication of course. We then have the canonical extension of by formal inverses of written (although I will use the notation to reinforce the conceptualization of a ring extension).
Now suppose is an -module. A natural question to ask is “can we induce some -module”? The natural choice will be the -module where we define
Proposition 1. Let be a ring and be a multiplicative subset. Then there exists a ring and homomorphism such that
- Elements of are invertible in
- For every homomorphism such that elements in are invertible in there exists a unique homomorphism such that
It naturally turns out that and for some Similarly we have:
Proposition 2. Let be a ring, be a multiplicative subset, and be an -module. Then there exists an -module and homomorphism such that
- For every the map defined by is bijective.
- For every -module such that the map is bijective for every and if we have a homomorphism there exists a unique homomorphism such that
Here we can show that and is called the ring of fractions of by while often denoted for short, is called the module of fractions of by
Definition 3. Let be an ideal in We define the radical of in as
We call the nilradical of is a reduced ring if
Let be an -algebra and be a multiplicative subset of Then since is an -module, we can make into an -module
But we can also turn it into an -algebra. We need a homomorphism
But the domain collapses to and then to under the product of So we have the canonical product we need.
 Bourbaki, N. Commutative Algebra, Ch 1-7. Springer-Verlag. 1989.