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