where is a indeterminate.
One may define localization in much more general situation. The reader is advised to read standard books on commutative algebras.
We need to show that it is injective, and that it gives a homeomorphism between and .
Let us do this by considering representations.
Now, for any , extends to if and only if the image of is invertible, that means, . In such a case, the extension is unique. (We recall the fact that the inverse of an element of a field is unique.)
It is easy to prove that is a homeomorphism.
Let be a ring. Let . It is important to note that each element of is written as a ``fraction''
One may introduce as a set of such formal fractions which satisfy ordinary computation rules. In precise, we have the following Lemma.
We introduce on the following equivalence law.
Then we may obtain a ring structure on by introducing the following sum and product.
where we have denoted by the equivalence class of .
Likewise, for any -module , we may define as a set of formal fractions
which satisfy certain computation rules.