where in the above notation is a indeterminate prepared for each element .) We denote by a canonical map .
Let be a ring, be a ring homomorphism such that is invertible in for any . Then there exists a unique ring homomorphism such that
Then (1) is an ideal of . Let us put , the canonical projection. Then:
(2) is multiplicatively closed.
(3) We have
(4) is injective.
where the equivalence relation is defined by
We may introduce a -module structure on in an obvious manner.
thus constructed satisfies an universality condition which the reader may easily guess.
We may also localize categories, but we need to deal with non commutativity of composition. To simplify the situation we only deal with a localization with some nice properties as follows:
In a simpler (but not rigorous) words, for each ``composable '', there exists such . Similarly, for each composable , there exists such that holds.