Commutative algebra - Department of Mathematical Sciences - old ...

58 4. FRACTION CONSTRUCTIONS
4.1.4. Proposition. Let U ⊂ R be a multiplicative subset.
(1) a
1 = 0 in U −1R if and only if Ann(a) ∩ U = ∅.
(2) The canonical ring homomorphism R → U −1R is injective if and only if U
consists only of nonzero divisors.
Proof. a
1 = 0 if and only if va = 0 for some v ∈ U.
4.1.5. Corollary. If R is a domain and U is the nonzero elements then K = U −1 R
is a field and the canonical homomorphism identifies R as a subring.
4.1.6. Definition. The field K in 4.1.5 is the fraction field of R.
4.1.7. Corollary. If R is a domain with fraction field K and φ : R → L is an
injective ring homomorphism into a field, then there is a unique homomorphism
K → L extending φ.
4.1.8. Definition. The total ring of fractions of R is U −1 R, where U is the set of
nonzero divisors of R.
4.1.9. Corollary. A ring is a subring of its total ring of fractions.
4.1.10. Proposition. Let U = {u n } the powers of an element u ∈ R. There is an
isomorphism
R[X]/(uX − 1) → U −1 R, X ↦→ 1
u
The ring of fractions with only one denominator is of finite type.
Proof. A pair of inverse homomorphism are constructed by 1.6.7 and 4.1.3.
4.1.11. Exercise. (1) Show that if U contains a nilpotent element, then U −1 R = 0.
(2) Show that
Ker R → U −1 R = {a ∈ R|ua = 0, for some u ∈ U}
(3) Let U = {u1, . . . , um} and u = u1 · · · um. Then show that
U −1 R = {u n } −1 R
(4) Let R1, R2 be domains with fraction fields K1, K2. Show that the total ring of fractions
of R1 × R2 is K1 × K2.
(5) Show that the ring
{ a
∈ Q|a ∈ Z, n ∈ N}
2n is of finite type over Z.
4.2. Modules of fractions
4.2.1. Lemma. Let R be a ring and U ⊂ R multiplicative. On U × M is defined
a relation
(u, x) ∼ (u ′ , x ′ ) ⇔ there is v ∈ U such that vu ′ x = vux ′
(1) The relation is an equivalence relation.
(2) If (u, x) ∼ (u ′ , x ′ ) and (v, y) ∼ (v ′ , y ′ ) then (uv, vx + uy) ∼ (u ′ v ′ , v ′ x ′ +
u ′ y ′ ).
(3) If (u, a) ∼ (u ′ , a ′ ) in U × R 4.1.1 and (v, x) ∼ (v ′ , x ′ ) then (uv, ax) ∼
(u ′ v ′ , a ′ x ′ ).
Proof. The claims are proved by simple calculations. See the proof 4.1.1.