5. Rings of Quotients and Localization

A multiplicatively closed subset of a ring is a set S ≠ ∅ such that st ∈ S whenever s, t ∈ S.

5.1 Definition. If S is a multiplicatively closed subset of a commutative ring R, then the ring of fractions with denominators in S, (S^{−1}R, +, ·) and the canonical homomorphism f_{S}: R → S^{−1}R are defined as follows:

S^{−1}R is the set of equivalence classes on S × R under the equivalence relation

(s, r) ∼ (s′, r′) :⟺ ∃t ∈ S tsr′ = ts′r.

The equivalence class of (s, r) is denoted by \frac{r}{s}. Addition and multiplication are defined by

\frac{r}{s} + \frac{r′}{s′} = \frac{s′r + sr′}{ss′}  and  \frac{r}{s} · \frac{r′}{s′} = \frac{rr′}{ss′}

and the canonical homomorphism f_{S}: R → S^{−1}R by f_{S}(r) = \frac{rs}{s} (for any s ∈ S).

A lot of facts have to be checked before the above definition can be called valid, for instance that ∼ is really an equivalence relation, that addition and multiplication are well-defined (i.e. the resulting equivalence class depends only in the equivalence classes of the arguments and not on the numerator and denominator of the fractions representing them), that S^{−1}R satisfies the ring axioms, and that f_{S} is well-defined and a ring homomorphism. These verifications are easy, if tedious. In doing them, one also stumbles upon the following properties of S^{−1}R:

(1) S^{−1}R is a commutative ring with zero \frac{0}{s} and unit element \frac{s}{s} (for any s ∈ S).

(2) \frac{r}{s} = \frac{r′}{s′} iff there exists a t ∈ S with trs′ = tr′s.

(3) If S contains no zero-divisors, then \frac{r}{s} = \frac{r′}{s′} iff rs′ = r′s.

(4) For all r ∈ R and s, t ∈ S \frac{r}{s} = \frac{rt}{st}.

(5) \frac{r}{s} = 0 iff there exists a t ∈ S with rt = 0.

(6) \frac{r}{s} is a unit iff there exists a t ∈ S with rt ∈ S.

(7) For every s ∈ S f_{S}(s) is a unit in S^{−1}R.

(8) Kerf_{S} = {r ∈ R | ∃t ∈ S rt = 0}.

(9) f_{S} is injective iff S contains no zero-divisor.

(10) If 0 ∈ S then S^{−1}R = {0}.

Recall that a multiplicatively closed set S is saturated, if S contains all divisors of every s ∈ S. The saturation of a multiplicatively closed set S is S̄ = {t ∈ R | ∃r ∈ R rt ∈ S}.

5.2 Lemma. Let S be a multiplicatively closed subset of R and S̄ its saturation. Then φ: S^{−1}R → S̄^{−1}R given by φ(\frac{r}{s}) = \frac{r}{s} is an isomorphism of rings.

Proof. φ is well-defined and a ringhomomorphism; to see injectivity, suppose \frac{r}{s} ∈ Kerφ, then rt = 0 for some t ∈ S̄. t divides some t′ ∈ S, such that t′r = 0 and therefore \frac{r}{s} = 0 in S^{−1}R. For surjectivity, consider \frac{r}{t} with r ∈ R and t ∈ S̄. For some s ∈ S and w ∈ R we have s = tw (and thus w ∈ S̄) and \frac{r}{t} = \frac{rw}{tw} = \frac{rw}{s} = φ(\frac{rw}{s}). □

By the previous lemma and Lemma 10.2, all rings of quotients of R are of the form form S^{−1}R with S = R \ ⋃_{P ∈P} P for a set P of prime ideals of R.

5.3 Definition. Important examples of rings of fractions:

• If S is the set of all non-zero-divisors of R, then S^{−1}R is called the complete ring of fractions of R; a special case of this is the quotient field of an integral domain, which is S^{−1}R with S = R \ {0}.

• If S = R \ P, P a prime ideal of R, then S^{−1}R is denoted R_{P} and called the localization of R at P. The ideal S^{−1}I of R_{P} is denoted I_{P} or IR_{P}.

The importance of the construction lies in the relationship between ideals of R and ideals of S^{−1}R. We already know that for every ideal I of R, f_{S}(I)S^{−1}R is an ideal of S^{−1}R - the extension of I into S^{−1}R. It is denoted by S^{−1}I and consists of all elements of S^{−1}R that admit a representation as a fraction with numerator in I:

5.4 Lemma.

S^{−1}I = \{\frac{i}{s} ∈ S^{−1}R | i ∈ I\}.

Note, however, that \frac{r}{s} ∈ S^{−1}I does not imply r ∈ I, exept in special cases such as:

5.5 Lemma. If P is a primary ideal of R with P ∩ S = ∅ then \frac{r}{s} ∈ S^{−1}P implies r ∈ P.

Likewise, for every ideal J of S^{−1}R, f_{S}^{−1}(J) is an ideal of R - the contraction of J into R. (If f_{S} is injective then f_{S}^{−1}(J) = J ∩ R.) Also, if J is a prime [or primary] ideal then so is f_{S}^{−1}(J). It is easy to see that f_{S}^{−1}(J) consists precisely of all numerators that occur in fraction representations of elements of J:

5.6 Lemma. Let J be an ideal of S^{−1}R. Then

f_{S}^{−1}(J) = \{r ∈ R | ∃s ∈ S \frac{r}{s} ∈ J\}.

In particular, for every ideal I of R, f_{S}^{−1}(S^{−1}I) is the set of numerators that occur in fractions representing elements of S^{−1}I.

5.7 Theorem.

(a) S^{−1}I = S^{−1}R iff I ∩ S ≠ ∅.

(b) For every ideal J of S^{−1}R, S^{−1}f_{S}^{−1}J = J; in particular, every ideal of S^{−1}R is of the form S^{−1}I for some ideal I of R.

(c) A bijection between the prime ideals P of R with S ∩ P = ∅ and all prime ideals of S^{−1}R is given by P ↦ S^{−1}P (for P a prime ideal of R), its inverse is Q ↦ f_{S}^{−1}Q (for Q a prime ideal of S^{−1}R).

(d) Like (c), with “prime” replaced by “primary”.

Rephrasing this in terms of extension and contraction, we have J^{ce} = J for every ideal J of S^{−1}R and P^{ec} = P for every primary ideal P of R with P ∩S = ∅.

5.8 Corollary. Let P be a prime ideal of R. Then every ideal of R_{P} is of the form I_{P} for some ideal I of R with I ⊆ P and via Q ↦ Q_{P} the prime ideals of R_{P} correspond bijectively to the prime ideals Q of R with Q ⊆ P. In particular, P_{P} is a maximal ideal of R_{P}, and its only maximal ideal.

By the above theorem, the localization of R at a prime ideal P contains a unique maximal ideal, P_{P}.

5.9 Definition. A ring containing exactly one maximal ideal is called a local ring. (Note, however, that many authors call such rings quasi-local, reserving the name local for Noetherian rings with a unique maximal ideal.)

5.10 Lemma. A ring is local if and only if the set of non-units forms an ideal.

Proof. Easy. □

5.11 Theorem(localization commutes with quotients). Let S be a multiplicative subset of R, I an ideal of R, and π: R → R/I the canonical projection then

(π(S))^{−1}(R/I) ≃ S^{−1}R/S^{−1}I.

5.12 Corollary. For every prime ideal P of R

((R/P) \ {0 + P })^{−1}R/P ≃ R_{P} /P_{P} .

In other words, if P is a prime ideal of R then the quotient field of the integral domain R/P is the same as the residue field of the maximal ideal P_{P} of R_{P}. This field is called the residue field of P.

∗ ∗ ∗ Symbolic powers ∗ ∗ ∗

5.13 Definition. Let P be a prime ideal of R. The n-th symbolic power of P is defined by P^{(n)} = R ∩ P^{n}R_{P}.

The n-th symbolic power of P is the contraction back to R of the extension of P^{n} into R_{P} and thus contains P^{n}.

5.14 Proposition. Let P be a prime ideal of R and n ∈ ℕ.

(1) P^{(n)} is P -primary.

(2) If Q is a primary ideal then P^{n} ⊆ Q ⟹ P^{(n)} ⊆ Q.

(3) P^{(n)} = P^{n} ⟺ P^{n} is primary.

5.15 Corollary. If M is a maximal ideal of R then M^{(n)} = M^{n} for all n ∈ ℕ.