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# Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 02.1   Let be a commutative ring. Let be its subset. We say that is multiplicative if
holds.

DEFINITION 02.2   Let be a multiplicative subset of a commutative ring . Then we define as

where in the above notation is a indeterminate prepared for each element .) We denote by a canonical map .

LEMMA 02.3   Let be a multiplicative subset of a commutative ring . Then the ring is characterized by the following property:

Let be a ring, be a ring homomorphism such that is invertible in for any . Then there exists a unique ring homomorphism such that

holds.

COROLLARY 02.4   Let be a multiplicative subset of a commutative ring . Let be an ideal of given by

such that

Then is an ideal of . Let us put , the canonical projection. Then:
1. is multiplicatively closed.

2. We have

3. is injective.

There is another description of . Namely, We consider an equivalence relateion on a set by

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 .

EXAMPLE 02.5   for . The total ring of quotients is defined as for

is not a zero divisor of A

When is an integral domain, then is the field of quotients of .

DEFINITION 02.6   Let be a commutative ring. Let be its prime ideal. Then we define the localization of with respect to by

DEFINITION 02.7   Let be a multiplicative subset of a commutative ring . Let be an -module we may define as

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.

PROPOSITION 02.8   Let be a commutative ring. Let be a multiplicative subet of . Let be an -module. Then we have an isomorphism

of -modules.

PROPOSITION 02.9   Let be a commutative ring. Let be a multiplicative subet of . Then the natural homomorphism is flat.

Next: local rings
2011-04-21