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

4
Fraction constructions
4.1. Rings of fractions
4.1.1. Lemma. Let R be a ring and U ⊂ R such that 1 ∈ U and for u, v ∈ U the
product uv ∈ U. On U × R is defined a relation
(u, a) ∼ (u ′ , a ′ ) ⇔ there is v ∈ U such that vu ′ a = vua ′
(1) The relation is an equivalence relation.
(2) If (u, a) ∼ (u ′ , a ′ ) and (v, b) ∼ (v ′ , b ′ ) then (uv, va + ub) ∼ (u ′ v ′ , v ′ a ′ +
u ′ b ′ ).
(3) If (u, a) ∼ (u ′ , a ′ ) and (v, b) ∼ (v ′ , b ′ ) then (uv, ab) ∼ (u ′ v ′ , a ′ b ′ ).
Proof. The claims are proved by simple calculations. (1) Symmetry is clear. Reflexive
follows as 1 ∈ U. Transitive: if (u, a) ∼ (u ′ , a ′ ), (u ′ , a ′ ) ∼ (u ′′ , a ′′ ) then
vu ′ a = vua ′ , v ′ u ′′ a ′ = v ′ u ′ a ′′ . Since multiplication is commutative (v ′ vu ′ )u ′′ a =
v ′ u ′′ vua ′ = (v ′ vu ′ )ua ′′ give (u, a) ∼ (u ′′ , a ′′ ). (2) From wu ′ a = wua ′ , w ′ v ′ b =
w ′ vb ′ follow that w ′ vv ′ wu ′ a = w ′ vv ′ wua ′ , wuu ′ w ′ v ′ b = wuu ′ w ′ vb ′ . So now
ww ′ (u ′ v ′ )(va + ub) = ww ′ (uv)(v ′ a ′ + u ′ b ′ ) as needed. (3) Is similar.
4.1.2. Definition. Let R be a ring. U ⊂ R is a multiplicative subset if 1 ∈ U
and for u, v ∈ U the product uv ∈ U. The ring of fractions U −1R is given by
on U × R under the relation 4.1.1
equivalence classes a
u
(u, a) ∼ (u ′ , a ′ ) ⇔ there is v ∈ U such that vu ′ a = vua ′
The addition is
a b
+
u v
and the multiplication is
a b
·
u v
The canonical ring homomorphism is
= va + ub
uv
= ab
uv
ι : R → U −1 R, a ↦→ a
1
4.1.3. Proposition. Let φ : R → S be a ring homomorphism and U ⊂ R a
multiplicative subset. If all elements in φ(U) ⊂ S are units, then there exists a
unique ring homomorphism φ ′ : U −1 R → S such that φ = φ ′ ◦ ι.
φ
R
ι
U −1R Proof. φ ′ ( a
u ) = φ(a)φ(u)−1 is the well defined unique ring homomorphism. Observe
that the elements of form bv −1 satisfy the rules for fractions.
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φ ′
S