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

62 4. FRACTION CONSTRUCTIONS
and get the exact sequence
Conclusion from 4.2.6.
0 → I → IU −1 R ∩ R → Ker(R/I → U −1 (R/I)) → 0
4.3.8. Corollary. Let P ⊂ R be a prime ideal. Then U −1 P ⊂ U −1 R is either a
prime ideal or the whole ring.
Proof. By 4.1.5 and 4.3.5 U −1 R/U −1 P = U −1 (R/P ) is either 0 or a domain.
4.3.9. Corollary. Let R be a principal ideal domain. Then U −1 R is a principal
ideal domain.
Proof. A restricted ideal is principal by hypothesis and the extension of a principal
ideal is principal. Conclude by 4.3.6.
4.3.10. Proposition. Let R be a unique factorization domain. Then U −1 R is a
unique factorization domain.
Proof. By 4.3.7 the extension of an irreducible element is either a unit or an irreducible
element. Since a principal ideal ( a a
u ) = ( 1 ), a factorization into irreducibles
in R gives a factorization in U −1R. Now if (a) = (p1) . . . (pn) is a factorization
in R and ( a
1 ) is irreducible in U −1R. Then all but one pi
1 is a unit in U −1R so
( a pi
1 ) = ( 1 ) is a prime ideal 4.3.9. The conditions 1.5.3 are satisfied.
4.3.11. Exercise. (1) Let U ⊂ R be multiplicative. Show that
U −1 (R[X]) = (U −1 R)[X]
(2) Let φ : R → S be a ring homomorphism and U ⊂ R a multiplicative subset. Show
that
U −1 S = φ(U) −1 S
4.4. Tensor modules of fractions
4.4.1. Proposition. Let R be a ring and U a multiplicative subset. For any module
M, the homomorphism
M ⊗R U −1 R → U −1 M, x ⊗ a ax
↦→
u u
is a natural isomorphism of U −1R-modules. Proof. By 2.6.3 there is an R-module homomorphism x ⊗ a
u
b a
ba bax b ax
(x ⊗ ) = x ⊗ ↦→ =
v
u
vu
vu
map U −1 M → M ⊗R U −1 R, x
u
v
ax → u . By definition
u , so the this is a U −1 R-homomorphism. The
↦→ x ⊗ 1
u
is an inverse.
4.4.2. Remark. The two constructions, module change of ring to a fraction ring
and fraction module are natural isomorphic functors from modules to modules over
the fraction ring.
4.4.3. Corollary. Let R be a ring and U a multiplicative subset. Then U −1 R is a
flat R-module.
Proof. This follows from 4.4.1 and 4.3.1.
4.4.4. Corollary. Let R be a ring and U a multiplicative subset and M, N modules.
Then there is a natural isomorphism
U −1 (M ⊗R N) U −1 M ⊗ U −1 R U −1 N