Appendix A Zorn’s Lemma and some Applications

Back

Zorn’s Lemma is a basic axiom of set theory; during our course in Higher

Algebra, we have had a number of occasions to use Zorn’s Lemma. Below,

I’ve tried to indicate exactly where we have made use of this important

axiom.

The setting for Zorn’s Lemma is a partially ordered set, which I now

define. If S is a set, and _ is a relation on S such that

(i) s _ s for all s 2 S,

(ii) if s1 _ s2 and s2 _ s1, then s1 = s2,

(iii) if s1 _ s2 and s2 _ s3, then s1 _ s3,

then (S,_) is called a partially ordered set. If (S,_) is a partially ordered

set such that whenever s1, s2 2 S we have s1 _ s2 or s2 _ s1, we call (S,_)

a totally ordered set. If (S,_) is a partially ordered set and if C is a totally

ordered subset of S, then C is called a chain. An upper bound for a chain

C _ S is an element s 2 S such that c _ s for all c 2 C. A maximal element

in the partially ordered set (S,_) is an element m 2 S such that if s 2 S

with m _ s, then m = s.

We are now ready to state Zorn’s Lemma:

Let (S,_) be a partially ordered set in which every chain in S has an

upper bound. Then S has a maximal element.

175

176 APPENDIX A. ZORN’S LEMMA AND SOME APPLICATIONS

We turn now to a few standard applications.

1. Basis of a Vector Space. Let V be a (possibly infinite dimensional)

vector space over the field F. We shall prove that V contains a basis, i.e.,

a linearly independent set which spans V . To prove this, let S be the set

of all linearly independent subsets of V , partially ordered by inclusion _.

Then (S,_) is a partially ordered set. Let C be a chain in S; to prove that

C has an upper bound, we construct the set

B =

[

A2C

A.

We can easily prove that B is linearly independent, which will show that B

is an upper bound for C. Indeed, suppose that b1, b2, . . . , br 2 B such that

there is a linear dependence relation of the form

Xr

i=1

_ibi = 0,

for some _1, _2, . . . , _r 2 F. Since C is a chain we see that for some A 2 C

we have b1, b2, . . . , br 2 A, which, of course, violates the fact that A is a

linearly independent subset of V . Thus, we can apply Zorn’s Lemma to

infer that there exists a maximal element M of S. We claim that M is a

basis of V . To prove this, we need only show that M spans V . But if there

is a vector v 2 V −span (M), thenM[{v} is a linearly independent subset

of V (i.e. is an element of S), which violates the maximality of M.

2. Maximal Ideals in Rings. Let R be a ring with identity 1. We can

apply Zorn’s Lemma to prove that R contains a proper maximal ideal M,

as follows. Let S = {proper ideals I _ R}, partially ordered by inclusion.

If C is a chain in S, form the set

J =

[

I2C

I.

Then it is easy to check that x, y 2 J implies that x + y 2 J, and that if

x 2 J, r 2 R, then rx, xr 2 J. Thus J is an ideal of R. Furthermore, it is

a proper ideal, for otherwise we would have 1 2 J, and so 1 2 I, for some

I 2 C, contrary to the assumption that I is a proper ideal of R. By Zorn’s

Lemma, we conclude that S has a maximal element M. It is then clear that

M is a maximal proper ideal of R.

177

3. Proof of Proposition 2.2.8 Let S be the set of ordered pairs (F_,  _)

such that F1 _ F_ and such that

F1 F2

F_ K2

-

-

6 6

 

 _

commutes. Partially order S by (F_,  _) _ (F_,  _) if and only if F_ _ F_

and  _|F_ =  _. Chains have upper bounds and so by Zorn’s Lemma there

is a maximal element ( ¯ F1, ¯  ). If ¯F 6= K1, then there exists f1(x) 2 F1 such

that f1(x) doesn’t split in ¯F1. Thus if ¯K1 is the splitting field over ¯F1 of

f1(x), then apply Proposition 2.2.7 to get

¯F

1 ¯  (¯F1),

¯K

1 ¯K2

-

-

6 6

¯ 

where ¯K2 is the splitting field for ˆ  (f1(x)) over ¯F1. This, of course, is a

contradiction to maximality.

4. Existence of an Algebraic Closure of a Given Field.

Lemma. If F is a field, then there exists an extension field F1 such that

every polynomial in F[x] has a root in F1 .

Proof. For each irreducible f = f(x) 2 F[x] let Xf be a corresponding

indeterminate, and set X = {Xf | f = f(x) 2 F[x] is irreducible }. We shall

work in the gigantic polynomial ring F[X] = F[. . . ,Xf , . . .]. Let I _ F[X]

be the ideal generated by the polynomials f(Xf ), where f = f(x) ranges

over the set of irreducible polynomials in F[x]. I claim that I 6= F[X].

For otherwise, there would exist polynomials f1(x), . . . fr(x) 2 F[x], and

polynomials g1, . . . gr 2 F[X] such that

1 = g1f1(Xf1) + g2f2(Xf2) + · · · + grfr(Xfr ).

178 APPENDIX A. ZORN’S LEMMA AND SOME APPLICATIONS

Let K be an extension field of F such that each fi(x) has a root _i 2 K, i =

1, 2, . . . , r. Let E : F[X] ! K[X] be the evaluation map that sends each

Xfi to _i, i = 1, 2, . . . , r, and maps all remaining Xh’s to themselves, where

h = h(x) 62 {f1(x), f2(x), . . . , fr(x)}. If we apply E to the above equation,

we get 1 = 0, a clear contradiction.

Next, form the quotient ring F[X]/I, which by the above, is not the

0-ring. By Zorn’s Lemma, there is a maximal ideal ¯M _ F[X]/I; if _ :

F[X] ! F[X]/I is the quotient map, then M := _−1( ¯M ) is a maximal ideal

of F[X]. Thus we have a field F1 := F[X]/M and an injection F ! F1. (As

usual, we can regard F as a subfield of F1.) Since each f(Xf ) 2 M, we see

that if f = Xf +M 2 F1, then f is a root of f(x) in F1. This proves the

lemma.

Proof of the Existence of Algebraic Closure. Let F = F0 be the

field whose algebraic closure we are to construct. By the above Lemma, we

may generate a sequence

F0 _ F1 _ F2 _ · · · ,

where every polynomial in Fi[x] has a root in Fi+1. Thus we may form the

field

E =

[

i_0

Fi;

clearly every polynomial f(x) 2 F[x] has a root in E. Thus if ¯F is the

subfield of E generated by the roots of all of the polynomials f(x) 2 F[x],

then clearly ¯F is an algebraic closure of F.

5. Free Modules over a Principal Ideal Domain.

Here we shall prove Proposition 5.3.11:

Proposition . Let M be a free module over the principal ideal domain R.

If N is a submodule of M, then N is free, and rank(N) _ rank(M).

Proof. We may certainly assume that N 6= 0; let B be a basis for M. For

any subset C _ B, set MC = R < C >, and set NC = N \MC.

Consider the set S of all triples (C0, C, f), where

(i) C0 _ C _ B,

(ii) NC is a free R-module,

179

(iii) f : C0 ! NC is a function such that f(C) is a basis of NC.

Since (;, ;, ;) 2 S, we see that S 6= ;. Now partially order S by

(C0, C, f) _ (D0,D, g) , C0 _ D0, C _ D and g|C0 = f.

It is easy to prove that chains have upper bounds and so Zorn’s lemma

guarantees a maximal element (A0,A, h) 2 S. By the above, we’ll be done

as soon as we show that A = B.

So assume that there is some b 2 B−A and set D = A[{b}. If ND = NA,

then clearly (A0,A, h) < (A0,D, h). Thus we may assume that ND properly

contains NA. Let I _ R be defined by setting

I = {r 2 R| y + rb 2 N, for some y 2 MA};

since ND properly contains NA, we have I 6= 0. Clearly I is an ideal of R.

Thus I = (s) for some s 2 R. We have w := x + sb 2 N for some x 2 MA.

Set D0 = A0 [ {b} and extend h0 : D0 ! ND by setting h0(b) = w. We shall

show that (D0,D, h0) 2 S.

We first show that h0(D0) spans ND. If z 2 ND then z = y + rb for

some r 2 R, y 2 MA. Also r = r0s for some r0 2 R and so z = y + r0sb =

y + r0(w − x) = (y − r0s) + r0w; also z − r0w = y − r0x 2 N \ MA = NA.

Therefore ND is spanned by h0(D0). Next, if h0(D0) is R-linearly dependent,

then {w} [ h0(A0) = {w} [ h(A0) is R-linearly dependent. Since h(A0) is

R-linearly independent, we infer that rw 2 R < h(A0) > \N = NA and so

rsb 2 MA which contradicts the fact that A[{b} is R-linearly independent.

The result follows.

As we mentioned in class, the only place we really used the above proposition

is in the proof of proposition 9, that is, in showing that finitely generated

torsion-free modules over the p.i.d. are free. Therefore, all we really need

is the above theorem in the case that M is finitely generated. The proof in

this case is quite simple, as indicated below.

First, a lemma. Note that we essentially proved this in class when we

proved proposition 10.

Lemma. If F is a free module over the ring R (not necessarily a p.i.d.),

and if _ : M ! F is an epimorphism of R-modules, then M _= ker(_) _ F

Proof. Let B be a basis of F, and for each b 2 B choose an element

b0 2 _−1(b). Now map B ! M by b 7! b0 thereby obtining a homorphism

180 APPENDIX A. ZORN’S LEMMA AND SOME APPLICATIONS

_ : F ! M which satisfies _ _ _ = 1F . Note that F _= __(M); thus it suffices

to show that M = ker(_)___(M). This is easy; recall how we did it in class.

Theorem. Let M be a finitely generated free module over the principal

ideal domain R. If N is a submodule of M, then N is free, and rank(N) _

rank(M).

Proof. We use induction on the rank of M. If the rank is 1, then, of course,

M _= R, in which case any submodule is just an ideal of R. Since R is a p.i.d.,

nonzero ideals are free, rank 1 submodules of R, so we’re done. Thus, assume

that the rank of M is greater than 1. Let {m1,m2, . . . ,mk} be a basis of

M, and let _ : M ! R be the homomorphism determined by _(m1) = . . . =

_(mk−1) = 0, _(mk) = 1. Note that ker(_) = R < m1, . . . ,mk−1 >, which is a

free R-module of rank k−1. Let N _ M be the given submodule of M; note

that ker(_ : N ! R) is a submodule of the free module R < m1, . . . ,mk−1 >.

By induction, we have that ker(_ : N ! R) is a free R-module of rank less

than or equal to k − 1. Since _(N) _ R is free, we apply the above lemma

to infer that N _= ker(_ : N ! R) _ _(N), and so N is a free R-module of

rank at most k.

6. The Equivalence of Divisible and Injective Abelian Groups.

Theorem. Let A be an abelian group. Then A is injective if and only if it

is divisible.

Proof. We shall first show that if A is injective, then it is divisible. Let

a 2 A and let d 2 Z. Consider the diagram

0 Z Z

A

- -

6

μd

_

Q

Q

Q Qk _

where _(1) = a. Let b = _(1). Then db = d_(1) = _(d) = _μd(1) = _(1) = a,

done.

Conversely, let A be divisible and consider the diagram

0 B0 B (exact).

A

- -

6

μ

_0

181

We may as well regard B0 _ B via μ. Let P = {(B00, _00)} such that

B0 _ B00 _ B and _00 : B00 ! A with _00|B0 = _0. Partially order by

(B00, _00) _ (C00, _00) if and only if B00 _ C00 and _00|B00 = _00. As (B0, _0) 2 P,

we see that P is nonempty. Clearly every chain in P has an upper bound

and so by Zorn’s Lemma, there exists a maximal element (B0

0, _0

0) 2 P. We

shall show that B0

0 = B. If not, then there exists b 2 B − B0

0; let m be the

order of the element b + B0

0 2 B/B0

0 . Set ˜B

00

= B0

0 + < b >.

Case 1: m = 1. Then < b > \B0

0 = 0 and so ˜B

00

= B0

0_ < b >, and

< b > is free. Then we can define _ :< b >! A arbitrarily and define

˜_0

0 : B0

0_ < b >! A by the universal property of _.

Case 2: m < 1. Now mb 2 B0

0 and _0

0(mb) 2 A. Find a 2 A with

ma = _0

0(mb) and define ˜_0

0 : ˜B

00

! A by setting ˜_0

0(b0

0 + nb) =

_0

0(b0

0)+na. One easily shows that ˜_0

0 is a well-defined homomorphism

which extends _0

0, so we are done.

7. Applications to Semisimple Modules

Lemma. A semisimple module has an irreducible submodule.

Proof. LetM be semisimple, and let 0 6= m 2 M. Let P = {submodules N _

M| m 62 N}. An easy application of Zorn’s lemma shows that P has a maximal

element M0. Since M is semisimple, there is a submodule M0 _ M such

that M = M0_M0; we shall show that M0 is irreducible. If not then M0 decomposes

as M0 = M0

1 _M0

2 where M0

1 ,M0

2 6= 0. But then, by maximality of

M0, we have m 2 M0_M0

1, M0_M0

2 and so m 2 M0_M0

1 \M0_M0

2 = M0,

a contradiction.

Theorem. The following conditions are equivalent for the R-module M.

(i) M is semisimple.

(ii) M =

P

i2I Mi, for some family {Mi| i 2 I} of irreducible submodules

of M.

(iii) M = _i2IMi, for some family {Mi| i 2 I} of irreducible submodules

of M.

Proof. (i))(ii): Let {M_| _ 2 A} be the set of all irreducible submodules

of M. We’ll show that M =

P

_2AM_. If not, then M =

P

_2AM__N for

182 APPENDIX A. ZORN’S LEMMA AND SOME APPLICATIONS

some submodule N. Apply the above lemma to conclude that N contains a

nonzero irreducible submodule of N, a clear contradiction.

(ii)) (iii): As above, let {M_| _ 2 A} be the set of all irreducible submodules

of M; by hypothesis, M =

P

M_. Let P = {B _ A|

P

L _2BM_ =

_2BM_} and partially order P by inclusion. Apply Zorn to get a maximal

element B0 _ A. Thus

P

_2B0

M_ =

L

_2B0

M_. If

P

_2B0

M_ 6=

M, then from

P

_2AM_ = M there must exist an irreducible submodule

M_ 6_

P

_2B0

M_. But then M_ \

P

_2B0

M_ = ;, i.e., M_ +

P

_2B0

M_ =

M_ _

P

_2B0

M_, contrary to the maximality of B0.

(iii))(i): Assume that {M_| _ 2 A} is the set of irreducibles in M;

thus M =

P

_2AM_. Let N _ M, and use Zorn’s lemma to obtain a

set C _ A which is maximal with respect to N \

P

_2C M_ = 0. If M 6=

N +

P

_2C M_, then there exists  2 A such that M 6_ N +

P

_2C M_. But

then M \(N +

P

_2C M_) = 0, and so N \(M +

P

_2C M_) = 0, contrary

to the maximality of C.

Index

k-transitively, 27

p-group, 6

principal ideal domain, 84

a.c.c., 135

action

imprimitive, 26

permutation, 7

primitive, 26

regular, 9

acts on, 5

adjoint

left, 180

right, 180

algebra, 170

algebraic, 45

algebraic closure, 50

algebraic integer, 95

algebraic integer domain, 96

algebraic number, 45

algebraically closed, 50

algorithm, 87

alternating, 177

alternating bilinear form, 35

alternating group, 23

ascending chain condition, 135

associates, 79

atomic domain, 85

automorphism, 8

balanced, 161

basis, 116

bilinear form

alternating, 35

bimodule, 165

Butterfly Lemma, 111

category theory, 180

Cauchy’s Theorem, 2

characteristic, 21, 30, 44

characteristic polynomial, 131

characteristic subgroup, 30

Chinese Remainder Theorem, 76

closed, 53

closure, 53

cofree R-module, 143

comaximal ideals, 76

commutative

graded algebra, 171

commutator, 30, 32

commutator subgroup, 30

companion matrix, 130

complete flag, 15

composite, 47

composition series, 31, 136

compositum, 47

conjugacy class, 5

content of a polynomial, 81

convolution, 152

coordinate mappinga, 115

cycle, 23

cycle type, 24

183

184 INDEX

cycles

disjoint, 23

cyclic R-module, 122

cyclic group, 3

cyclotomic polynomial, 69

d.c.c, 135

Dedekind Domain, 100

Dedekind Independence Lemma, 52

degree

of an element, 45, 174

of an extension, 44, 45

degree of an extension, 44

descending chain condition, 135

determinantal rank, 126

differential, 119

dihedral group, 4

direct sum, 113

discrete valuation ring, 108

discriminant, 65

disjoint cycles, 23

divides, 79

divisible abelian group, 142

division algorithm, 87

division ring, 139

double transitivity, 9

elementary components, 123

elementary divisors, 123

equivariant mapping, 7

Euclidean domain, 87

exact, 33, 92

exponent, 122

extension, 44

Galois, 54

purely inseparable, 58

separable, 58

simple, 45, 73

extension degree, 44

exterior algebra, 175

F-algebra, 152

field extension, 44

finitely generated, 84

submodule, 92

fixed point set, 5

fixed points, 5

flag, 15

type, 15

fractional ideal, 104

principal fractional ideal, 104

Frattini subgroup, 36

free

group, 38

module, 116

free R-module, 145

free product, 42

Frobenius automorphism, 60

Fundamental Theorem of Algebra,

65

Fundamental Theorem of Algebraic

Number Theory, 102

Fundamental Theorem of Galois Theory,

54

Galois extension, 54

Galois group, 52

general linear group, 14

generalized quaternion group, 20

generator

of a group, 3

generators and relations, 39

graded

algebra, 174

ideal, 174

graded algebra, 171

graded-commutative, 171

Grassmann space, 178

INDEX 185

greatest common divisor, 79, 80

group action, 5

faithful, 5

group algebra, 152

group of units, 79, 181

group ring, 152

Heisenberg Group, 35

Hilbert Basis Theorem, 84

homogeneous

elements, 174

homogeneous ideal, 174

homomorphism

module, 92

Hopkin’s Theorem, 158

ideal class group, 104

idempotent, 149

idempotents

orthogonal, 149

imprimitivity, 26

imprivitively, 26

Inclusion-Exclusion Principle, 62

integral domain, 75

integral group ring, 181

integrally closed, 96

internal direct sum, 92, 94, 114

invariant basis number, 117

invariant basis number (IBN), 117

invariant factors, 123

invertible ideal, 106, 142

involution, 4

irreducible, 79, 136

R-module, 136

R-module, 149

Jacobson radical, 155

Jordan canonical form, 132

Jordan-H¨older Theorem, 31

kernel of the action, 5

Kronecker product, 166

Krull topology, 64

Lagrange’s Theorem, 1

least common multiple, 79, 80

left adjoint, 180

left Artinian, 157

left Noetherian, 157

local ring, 138

localization, 107

lower central series, 32

maximal ideal, 75

minimal polynomial, 45

of a linear transformation, 129

minor, 126

modular law, 93, 111

module, 91

module homomorphism, 92

monomorphism, 181

Nakayama’s Lemma, 157

near field, 63

nil ideal, 156

nilpotent

element, 156

group, 32

ideal, 156

nilpotent element, 78

Noether Isomorphism Theorem, 111

Noetherian

module, 93

ring, 84

Noetherian module, 135

norm map, 61

normal closure, 39

normal series, 31

orbit, 5

186 INDEX

Orbit-Stabilizer Reciprocity Theorem,

5

order, 2, 121

infinite, 2

overring, 107

p-part, 12

perfect, 59

permutation isomorphic, 7

Pl¨ucker embedding, 178

pointwise, 152

polynomial

separable, 58

Primary Decomposition Theorem,

132

primary ideal, 77

prime

element, 79

ideal, 75

primitive, 26

primitive element, 73

Primitive Element Theorem, 73

primitive polynomial, 81

principal ideal, 76

prinicpal fractional ideal, 104

projection mappings, 115

projective, 141

projective general linear group, 14

projective space, 15, 178

projective special linear group, 14

purely inseparable, 58

element, 58

quadratic integer domains, 96

quasi-dihedral group, 20

quaternion group, 20

generalized, 20

rank, 117

rational canonical form, 130

regular action, 9

regular normal subgroup, 28

relations, 39

relations matrix, 125

relatively prime ideals, 76

representation, 152

residual quotient, 77

right adjoint, 180

right Artinian, 157

right Noetherian, 157

root tower, 72

Schreier Refinement Theorem, 136

Second Isomorphism Theorem, 111

semi-direct product, 17

external, 18

internal, 17

semidihedral group, 20

semisimple, 133

linear transformation, 139

R-module, 148

ring, 157

separable, 58, 73

element, 58

extension, 58, 73

polynomial, 58

separable element, 73

short exact sequence, 92

splitting, 118

splitting of, 95

simple, 136

simple R-module, 136

simple field extension, 45

simple radical extension, 72

simple ring, 150

Smith equivalent, 125

solvable, 31

group, 31

solvable by radicals, 72

INDEX 187

special linear group, 14

split short exact sequence, 118

splits, 95

splitting field, 46, 49

stabilizer, 5

stable, 54

subgroup

characteristic, 21, 30

submultiplicative algorithm, 87

subnormal series, 31

Sylow subgroup, 12

symmetric algebra, 174

symmetric group, 3

system of imprimitivity, 26

non-trivial, 26

trivial, 26

tensor algebra, 173

tensor product, 161

Third Isomorphism Theorem, 111

torsion element, 121

torsion submodule, 121

torsion-free, 121

totally discontinuous, 64

transitive, 7

transposition, 23

u.f.d., 79

unique factorization domain, 79

unit, 79

valuation ring, 108

word problem, 40

Zassenhaus Lemma, 111

zero-divisor, 75