**Yoshifumi Tsuchimoto**

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
*

such that

*(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:

- .

- Let
. Let
,
. Then there exist
and
morphisms
, and
such that
the diagram
In a simpler (but not rigorous) words, for each ``composable '', there exists such . Similarly, for each composable , there exists such that holds.

- Let
,
. Then the
following conditions are equivalent:
- There exists and such that .
- There exists and such that .

- If
and if
then
.