for some .
By treating as a fraction , construct the structure of a ring on the set .
This ring is called the localization of at and is denoted by (other common notation: ).
What does this mean?
This is a way of “building” inverses for a certain subset.
For example, say some element of has no inverse (it is not a unit in ).
Then in then the element is now the inverse, and now is a unit in this “bigger” ring.
Properties
There is a canonical ring homomorphism , .
Proof
is also a ring homomorphism.
We have (using and as the addition and multiplication in ),
and
for .
Universal Property
is characterized by the following universal property: for all ring homomorphisms for which each element of is sent to a unit of , there exists a unique ring homomorphism which makes the diagram
commute.
Proof
Let be a function as described above. Consider the candidate,
Note that makes sense since every element of maps to a unit in .
Let , using the fact that is a ring homomorphism, we have,
That is, is a ring homomorphism.
Using the canonical homomorphism from part (a), we have that for any element ,
Thus, it remains to show that is unique.
Let be another function that satisfies the universal property.
We know that by definition , so it is only necessary to check that .