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 is an ideal of . Let us put , the canonical projection. Then:
We call the quotient space space as . The equivalence class of in is denoted by . Then it is easy to introduce a ring structure of and see that actually satisfies the universal property of . We thus have a canonical isomorphism .
When is an integral domain, then is the field of quotients of .
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.
By a universality argument, we may easily see the following proposition.