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```                            Contents
Prerequisites
1. A dictionary on rings and ideals
1.1. Rings
1.2. Ideals
1.3. Prime ideals
1.4. Chinese remainders
1.5. Unique factorization
1.6. Polynomials
1.7. Roots
1.8. Fields
1.9. Power series
2. Modules
2.1. Modules and homomorphisms
2.2. Submodules and factor modules
2.3. Kernel and cokernel
2.4. Sum and product
2.5. Homomorphism modules
2.6. Tensor product modules
2.7. Change of rings
3. Exact sequences of modules
3.1. Exact sequences
3.2. The snake lemma
3.3. Exactness of Hom
3.4. Exactness of tensor
3.5. Projective modules
3.6. Injective modules
3.7. Flat modules
4. Fraction constructions
4.1. Rings of fractions
4.2. Modules of fractions
4.3. Exactness of fractions
4.4. Tensor modules of fractions
4.5. Homomorphism modules of fractions
4.6. The polynomial ring is factorial
5. Localization
5.1. Prime ideals
5.2. Localization of rings
5.3. Localization of modules
5.4. The local-global principle
5.5. Flat ring homomorphisms
5.6. Faithfully flat ring homomorphisms
6. Finite modules
6.1. Finite modules
6.2. Free modules
6.3. Cayley-Hamilton's theorem
6.4. Nakayama's lemma
6.5. Finite presented modules
6.6. Finite ring homomorphisms
7. Modules of finite length
7.1. Simple modules
7.2. The length
7.3. Artinian modules
7.4. Artinian rings
7.5. Localization
7.6. Local artinian ring
8. Noetherian rings
8.1. Noetherian modules
8.2. Noetherian rings
8.3. Finite type rings
8.4. Power series rings
8.5. Localization of noetherian rings
8.6. Prime filtrations of modules
9. Primary decomposition
9.1. Zariski topology
9.2. Support of modules
9.3. Ass of modules
9.4. Primary modules
9.5. Decomposition of modules
9.6. Decomposition of ideals
10. Dedekind rings
10.1. Principal ideal domains
10.2. Discrete valuation rings
10.3. Dedekind domains
Bibliography
Index
```
##### Document Text Contents
Page 1

ELEMENTARY
COMMUTATIVE ALGEBRA

LECTURE NOTES

H.A. NIELSEN

DEPARTMENT OF MATHEMATICAL SCIENCES
UNIVERSITY OF AARHUS

2005

Page 75

5.2. LOCALIZATION OF RINGS 75

Proof. (1)⇒ (2): A prime ideal P 6= 0 contains an irreducible element generating
a prime ideal. (2)⇒ (1): LetU be the multiplicative subset generated by generators
of principal prime ideals. Given a ∈ R nonzero and not a unit. If a /∈ U then by
5.1.6 there is a prime ideal (a) ⊂ P such that P ∩ U = ∅. Such a P contains no
principal prime ideal, so a ∈ U and 1.5.3 are satisfied.

5.1.13. Exercise. (1) Let K ⊂ R be an infinite subfield and I, P1, . . . , Pn any ideals.
Show that if I ⊂ P1 ∪ · · · ∪ Pn then I ⊂ Pi for some i.

(2) Let P, P1, P2 be proper ideals. Show that if P is a maximal ideal and Pn ⊂ P1 ∪P2
then P = P1 of P = P2.

5.2. Localization of rings

5.2.1. Definition. A ring R which contains precisely one maximal ideal P is a
local ring and denoted (R,P ). The residue field of R is R/P denoted by k(P ).
A ring homomorphism φ : R → S of local rings (R,P ), (S,Q) is a local ring
homomorphism if φ(P ) ⊂ Q.
5.2.2. Proposition. Let R be a ring.

(1) If (R,P ) is a local ring, then R\P is the set of units of R.
(2) If the subset of non units in R is an ideal P , then (R,P ) is a local ring.

Proof. (1) If u /∈ P then by 5.1.1 (u) = R and u is a unit. (2) Any ideal I 6= R
contains only non units, so I ⊂ P .

5.2.3. Proposition. A ring homomorphism φ : R→ S of local rings (R,P ), (S,Q)
is a local ring homomorphism if the extended ideal PS ⊂ Q or the contracted ideal
Q ∩R = P . The residue homomorphism k(P )→ k(Q) is a field extension.

Proof. The contraction Q ∩R is a prime ideal containing P . The rest is clear.

5.2.4. Lemma. Let R be a ring and P a prime ideal. Then U = R\P is a multi-
plicative subset. The ring of fractions U−1R is a local ring. The maximal ideal is
the extended ideal PU−1R. The residue field is U−1R/PU−1R which is canoni-
cal isomorphic to the fraction field of R/P .

5.2.5. Definition. Let R be a ring, P a prime ideal and U = R\P . The localized
ring at P is the local ring RP = U−1R. The residue field is denoted k(P ) =
RP /PRP .

R

��

// RP

��
R/P // k(P )

Note that

k(P ) = RP /PRP = (R/P )P = (R/P )(0)
5.2.6. Proposition. Given a ring homomorphism R → S and a prime ideal Q ⊂
S. Then P = Q ∩ R is a prime ideal and there is a local ring homomorphism

Page 76

76 5. LOCALIZATION

(RP , PRP )→ (SQ, QSQ) fitting into the following commutative diagram.

S

��

// SQ

��

R

��

//

55kkkkkkkkkkkkkkkkkkkk
RP

��

55kkkkkkkkkkkkkkkkkk

S/Q // k(Q)

R/P //

55lllllllllllllllll
k(P )

55lllllllllllllllll

Proof. This is clear from the constructions.

5.2.7. Example. Let the ring be Z.
(1) The local ring at (0) is the fraction field Q = Z(0).
(2) The local ring Z(p) for a prime number p is identified with a subring of Q

Z(p) = {
m

n
|p not dividing n}

The residue field Fp = Z(p)/(p).
(3) Any nonzero ideal in Z(p) is principal of the form (p

n) for some n.

5.2.8. Proposition. Let (R,P ) be a local ring. One of the following conditions is
satisfied:
(1) The characteristic char(R) = 0. P ∩ Z = (0) and Q ⊂ R is a subfield.

Q→ k(P ) is a field extension.
(2) The characteristic char(R) = 0. P ∩ Z = (p), p a prime number. Z(p) ⊂ R

is a local subring. Fp → k(P ) is a field extension.
(3) The characteristic char(R) = pn, a power of a prime number. Z/(pn) ⊂ R

is a local subring. Fp → k(P ) is a field extension.

Proof. (1) (2) are clear by 5.2.3 and 5.2.7. (3) If the characteristic is nonzero then
any prime ideal contracts Q ∩ Z = (p). So a prime number q 6= p gives a unit in
R. There is a local homomorphism Z(p) → R, 4.1.3. The nontrivial kernel is (pn)
by 5.2.7.

5.2.9. Example. (1) A field K is a local ring with maximal ideal (0). The power
series ringK[[X]] is a local ring with maximal ideal (X) and residue fieldK.

(2) Let (R,P ) be a local ring. The power series ring R[[X]] is a local ring with
maximal ideal (P,X) and residue field k(P,X) = k(P ).

5.2.10. Proposition. Let R be a domain.
(1) The local ring at (0) is the fraction field K = R(0).
(2) Any local ring RP is identified with a subring of the fraction field K.
(3) The intersection

R =

P

RP , P a maximal ideal

5.2.11. Proposition. Let R× S be a product of rings.

Page 149

INDEX 149

injective modules, 56
irreducible component, 125
irreducible element, 15
irreducible principal ideal, 15
irreducible space, 125
irreducible subset, 125
isomorphism, 9, 21

kernel, 11, 25
kernel and cokernel, 25
Krull’s intersection theorem, 117, 120
Krull’s theorem, 73

least common multiple, 16
left exact contravariant functor, 50
left exact functor, 50
length, 102
linear map, 22
local artinian ring, 110
local parameter, 141
local ring, 75
local ring homomorphism, 75
localization, 109
localization of modules, 77
localization of noetherian rings, 120
localization of rings, 75
localized homomorphism, 77
localized module, 77
localized ring, 75
locally free module, 79

maximal ideal, 13
minimal prime, 126
minimal prime ideal, 74
minor, 89
module, 21
module of fractions, 65
modules and homomorphisms, 21
modules of fractions, 65
monic polynomial, 16
monomial, 16
multilinear:, 89
multiplication, 9
multiplication of principal ideals, 15
multiplicative subset, 63
multiplicity, 18

Nakayama’s lemma, 93
natural homomorphism, 31
natural isomorphism, 31
negative, 9
nilpotent, 14
noetherian module, 113
noetherian ring, 115
noetherian rings, 115

noncommutative ring, 9
nontrivial idempotent, 15
nonzero divisor, 10, 22
normed:, 90
notherian modules, 113

order, 20

polynomial, 16
polynomial ring, 16
polynomials, 16
power series, 20
power series ring, 20
power series rings, 118
primary decomposition, 134
primary ideal, 132
primary modules, 132
primary submodule, 132
prime divisor, 15
prime element, 15
prime fields, 19
prime filtrations of modules, 121
prime ideal, 13
prime ideals, 13, 73
principal ideal, 11
principal ideal domain, 15
principal ideal domains, 139
principal open subsets, 123
product ring, 10
projection, 9, 24
projections, 28
projective module, 55
projective modules, 55
proper ideal, 11

rank, 91
reduced, 14
reduced primary decomposition, 134
reflexive module, 33
residue field, 75
residue homomorphism, 77
restriction of scalars, 22
retraction, 43
right exact contravariant functor , 50
right exact functor, 50
ring, 9
ring extension, 9
ring generated, 17
ring of fractions, 63
rings, 9
rings of fractions, 63
root, 18
roots, 18

scalar multiplication, 21, 22
section, 43
semi-local ring, 128

Page 150

150 INDEX

short exact sequence, 43
simple module, 101
simple modules, 101
simple root, 18
snake homomorphism, 46
snake lemma, 47
spectrum, 123
split exact sequence, 44
standard basis, 29
subfield, 19
subgroup, 9
submodule, 21
submodule generated, 23
submodules and factor modules, 23
subring, 9
sum and product, 28
support, 126
support of modules, 126
symbolic power, 136

tensor modules of fractions, 69
tensor product, 34
tensor product modules, 34
tensor product ring, 39
the length, 102
The local-global principle, 79
the polynomial ring is factorial, 71
the snake lemma, 45
torsion element, 139
torsion free module, 139
torsion module, 139
torsion submodule, 139
total ring of fractions, 64

uniformizing parameter, 141
unique factorization, 15
unique factorization domain, 15
unit, 10

valuation, 142
vector space, 22

windmill lemma, 49

Zariski topology, 123
zero, 9
zero divisor, 10, 22
zero ideal, 11
zero module, 21
zero submodule, 21