A. C. C. see ascending
chain condition Abel's Theorem 308
abelian field extension 292
abelian group 24
—defined by generators
and relations 343ff divisible— 195ff finitely generated— 76ff free— 7 Iff action of group on set 88ff additive notation for
groups 25, 29, 38, 70
adjoining a root 236
adjoint —associativity 214
classical—matrix 353
—pair of functors 477ff affine variety 409
algebra(s) 226ff, 450ff division— 456ff Fundamental Theorem
of— 266
group— 227
homomorphism of—
228
—ideal 228
left Artinian— 451
tensor product of— 229
algebraic —algebra 453
—closure 259ff —element 233, 453
—field extension 233ff —number 242
—set 409
algebraically —closed field 258
—{independent 31 Iff algorithm division— 11, 158
Euclidean— 141
alternating —group 49
—multilinear function 349
annihilator 206, 388,417
anti-isomorphism 330
antisymmetric relation 13
Aitin, E. vi, 251,252, 288
Artinian left—algebras 451
—module 372fT
—rings 372ff, 419ff, 435ff ascending —central series 100
—chain condition 83, 372ff associate 135
associated prime ideal 380,
387
associative —binary operation 24
generalized—law 28
AutkF 243
automorphism 30, 119
extendible— 251, 260
inner—91,459
A^-automorphism 243
—of a group 30, 90
axiom —of choice 13
—of class formation 2
—of extensionality 2
—of pair formation 6
power— 3
Baer lower radical 444ff base
transcendence— 313ff basis 71, 181
dual— 204
Hilbert—Theorem 391
matrix relative to a—
329ff standard— 336
transcendence— 313ff
bijective function 5
bilinear map 211, 349
canonical— 211
bimodule 202
binary operation 24
closed under a— 31
binomial —coefficient 118
—theorem 118
Boolean ring 120
bound (greatest) lower— 14
least upper— 14
upper— 13
cancellation 24, 116
canonical —bilinear map 211
—epimorphism 43, 125,
172
—forms 337
—injection 60, 130, 173
Jordan—form 361
—middle linear map 209
primary rational—form 361
—projection 59,130,173
rational—form 360
Cardan's formulas 310
cardinal number(s) 16ff exponentiation of— 22
—inequalities 17
product of— 16
sum of— 16
trichotomy law for— 18
cardinality 16
Cartesian product 6, 7ff category 52ff, 464ff concrete 55
dual— 466
opposite— 466
product— 467
Cauchy's Theorem 93
Cayley's Theorem 90
Cayley-Hamilton Theorem
367, 369
center —of a group 34, 91
—of a ring 122
central ascending—series 100
—idempotents 135
—simple division algebra 456
centralizer 89
chain 13
chain conditions 83ff, 372ff char/? 119
characteristic —of a ring 119
—polynomial 366fF
—vectors 367
—values 367
—space 368
—subgroups 103
Chinese Remainder
Theorem 131
choice axiom of— 13
—function 15
class 2ff axiom of—formation 2
—equation 90, 91
equivalence— 6
proper— 2
quotient— 6
classical adjoint 353
classification of finite
groups 96ff closed
—intermediate field 247
—object 410
—subgroups 247
—under a binary operation 31
closure
algebraic— 259fT
integral— 397
normal— 265
codomain 3
coequalizer 482
cofactor 352
Cohen, I. S. 388
coimage 176
cokernel 176
difference— 482
—of a morphism 483
column elementary—operation
338
—rank 336
—space 336
—vector 333
commutative —binary operation 24
—diagram 4
generalized—law 28
—group 24
—ring 115
commutator 102
—subgroup 102
companion matrix 358
comparable elements 13
complement 3
complete —direct sum 59
—lattice 15
—ring of quotients 144
—set of coset representatives 39
completely reducible
module 437
composite —function 4
—functor 468
—morphism 52
—subfield 233
composition series 108, 375
concrete category 55
congruence —modulo m 12
—modulo a subgroup 37
—relation 27
conjugacy class 89
conjugate elements or
subgroups 89
conjugation 88
constant —polynomial 150
—term 150, 154
constructible number 239
content of a polynomial 162
Continuum Hypothesis 17
contraction of an ideal 146, 398
contravariant —functor 466
—horn functor 466
coordinate ring 413
coproduct 54ff correspondence Galois— 246
coset 38ff
complete set of—
representatives 39
couniversal object 57
covariant —functor 465
—horn functor 465
Cramer's Rule 354
cubic
résolvant— 272
cycles 46
disjoint— 47
Conway, C. T., xiii
cyclic
—field extension 289fT
—group 32, 35ff —module 171
—subspace 356
cyclotomic
—extension 297ÎT
—polynomial 166, 298
D. C. C—, see descending
chain condition Dedekind domain 401 ff degree —of a field element 234
—of inseparability 285ff —of a polynomial 157, 158
transcendence— 316
delta
Kronecker— 204
De M organ's Laws 3
dense ring 418
Density Theorem 420
denumerable set 16
dependence algebraic— 311 ff linear— 181
derivative 161
derived subgroup 102
descending chain condition
83, 372ff determinant 35Iff expansion of a—along a
row 353
—of an endomorphism 354
diagonal matrix 328
diagram commutative— 4
difference —kernel 482
—cokemel 482
dihedral group 50
dimension 185ff relative— 245
invariant—property 185
direct —factor 63
—product 26, 59ff, 130,
131, 173
—sum 60, 62ff, 173, 175
—sum of matrices 360
—summand 63, 437
discriminant 270
discrete valuation ring 404
disjoint linearly—subfields 318ff —permutations 47
—sets 3
—union 58
divisible group 195ff division —algebras 227, 456ff —algorithm 11, 158
finite—ring 462
—ring 116
divisor(s) elementary— 80, 225,
357, 361
greatest common— 11,
140
zero— 116
domain
Euclidean— 139
integral— 116
—of a function 3
principal ideal— 123
Prufer— 409
valuation— 409
unique factorization—
137
double dual 205
dual —basis 204
—category 466
double— 205
—map 204
—module 203ff —statement 54
echelon reduced row—form 346
eigenspace 368
eigenvalues 367ff
eigenvectors 367ff Eisenstein's Criterion 164
elementary —column operation 338
—divisors 80, 225, 357, 361
—Jordan matrix 359
—row operation 338
—symmetric functions 252
—transformation matrix 338
embedded prime 384
embedding 119
empty —set 3
—word 64
End A 116
endomorphism(s)
characteristic
polynomial of an—
366ff
dense ring of— 418
determinant of an— 354
matrix of an— 329ff, 333
nilpotent— 84
normal— 85
—of a group 30
—ring 116, 179,415
trace of an— 369
epic morphism 480
epimorphism, 30, 118, 170
canonical— 43, 125, 172
—in a category 480
equality 2
equalizer 482
equation(s)
general—of degree n
308ff linear— 346
equipollent sets 15
equivalence —class 6
—in a category 53
natural— 468
—relation 6
equivalent
—matrices 332, 335fF
—series 109, 375
Euclidean —algorithm 141
—ring 139
Euler function 297
evaluation homomorphism 153
even permutation 48
exact sequence 175ff short— 176
split— 177
exponentiation —in a group 28
—of cardinal numbers 22
extendible automorphism
251,260
extension field 23 Iff abelian— 292
algebraic— 233ff cyclic— 289ff, 292
cyclotomic— 297ff finitely generated— 231
Galois— 245
normal— 264ff
purely inseparable—
282ff
purely transcendental—
314
radical— 303ff separable— 261,282ff, 324
separably generated—
322
simple— 232
transcendental— 233, 311ff extension integral— 395
—of an ideal 145
—ring 394ff extensionality
axiom of— 2
external —direct sum 60, 173
—direct product 130
—weak direct product 60
factor group 42ff
factorization
—in commutative rings
135ff
—in polynomial rings
157ff
unique" factorization—
137ff factors invariant—80, 225, 357, 361
—of a series 107, 375
faithful module 418
field 116, 230ff
algebraically closed—
258
—extension 23Iff finite— 278ff, 462
fixed— 245
intermediate— 231
—of rational functions
233
perfect— 289
splitting— 257ff finite dimensional —algebra 227
—field extension 231
—vector space 186
finite field 278ff, 462
finite group 24, 92ff, 96ff finite set 16
finitely generated —abelian groups 76ff, 343ff
—extension field 232
—group 32
—ideal 123
—module 171
first isomorphism theorem
44,126,172
Fitting's Lemma 84
five lemma 180
short— 176
fixed field 245
forgetful functor 466
form
Jordan canonical—361
multilinear— 349ff primary rational canoni- cal— 361
rational canonical— 360
formal —derivative 161
—power series 154ff four group 29
fractional ideal 401
free
—abelian group 7Iff —group 65ff —module 18 Iff, 188
—object functor 479
—object in a category
55ff —product 68
rank or dimension of a—
module 185
freshman's dream 121
Frobenius Theorem 461
fully invariant subgroup 103
function(s) 3ff bijective— 5
bilinear—211, 349
choice— 15
field of rational— 233
graph of a— 6
injective— 4
left or right inverse of a
— 5
Moebius— 301
multilinear— 349ff surjective— 5
symmetric rational—
252ff
two sided inverse of a—
5
functor 465ff adjoint— 477ff composite— 468
forgetful— 466
free object— 479
horn— 465, 466
representable— 470ff Fundamental Theorem —of Algebra 265ff —of Arithmetic 12
—of Galois Theory 245ff, 263
g.l.b. 14
Galois —correspondence 246
—extension 245
—fields 278ff Fundamental Theorem of—Theory 245ff, 263
—groups 243
—group of a polynomial
269ff —Theory 230ff Gauss Lemma 162
Gaussian Integers 139
general —equation of degree n 308ff
—polynomial of degree n 308
generalized —associative law 28
—commutative law 28
—fundamental theorem of Galois Theory 263
—multiplicative system 449
generators and relations
67ff, 343ff generators —of a group 32
—of an ideal 123
—of a module 171
Going-up Theorem 399
Goldie —ring 447ff —'s Theorem 448
Goldmann ring 400
graph 6
greatest —common divisor 11, 140
—lower bound 14
group(s) 24ff abelian— 24, 70if, 76ff —algebra 227
alternating— 49
center of a— 34, 91
cyclic— 32, 35fF
—defined by generators and relations 67, 343flf dihedral— 50
direct product of— 26, 59ff
direct sum of— 60, 62
divisible— 195ff factor— 42
finite— 92ff
finitely generated— 32,
76ff free— 65ff free abelian— 7Iff free product of— 68
Galois— 243
Galois—of a polynomial
269ff
homomorphism of—
30ff, 43ff
identity element of a—
24
indecomposable— 83
inverses in— 24
isomorphic— 30
isotropy— 89
join of— 33
metacyclic— 99
nilpotent— 100
—of integers modulo m, 27
—of rationals modulo
one 27
—of small order 98
-of symmetries of the
square 26
/7-group 93
quaternion— 33, 97, 99
reduced— 198
—ring 117
simple— 49
solvable— 102ff, 108
symmetric— 26, 46ff torsion— 78
weak direct product of—
60, 62
Hall's Theorem 104
Halmos, P. ix -Hamilton Cay ley—Theorem 367, 369
Hilbert —Basis Theorem 391
—Nullstellensatz 409ff, 412
—'s Theorem 90 292
-Holder Jordan—Theorem 111, 375
Hom/K/4,#) 174, 199ff horn functor 465, 466
homogeneous —of degree k 158
—system of equations 346
homorphism induced— 199, 209
^-homomorphism 243
kernel of a— 31, 119, 170
matrix of a— 329ff, 333
natural— 469
—of algebras 228
—of groups 30flf, 43fF
—of modules 170ff, 199ff —of rings 118
substitution— 153
Hopkin's Theorem 443
ideal(s) 122ff algebra— 228
finitely generated— 123
fractional— 401
—generated by a set 123
invertible— 401
left quasi-regular— 426
maximal— 127ff nil— 430
nilpotent— 430
order— 220
primary— 380ff prime— 126ff, 377ff primitive— 425
principal—ring 123
proper— 123
regular left— 417
idempotent —element 135,437
orthogonal—s 135, 439
identity 24, 115
—function 4
—functor 465
—matrix 328
image inverse— 4, 31
—of a function 4
—of a homomorphism
31, 119,170
—of a set 4
inclusion map 4
indecomposable group 83
independence algebraic— 31 Iff linear— 74, 181
independent
set of ideals 446
indeterminate 150, 152
index —of a subgroup 38
relative— 245
induced homomorphism
199 209
induction mathematical— 10
transfinite— 14
infinite set 16
initial object 57
injections
canonical— 60, 130, 173
injective —function 4
—module 193ff inner automorphism 91,
459
inseparable —degree 285ff purely—extension 282ff integers 9ff Gaussian— 139
—modulo m 27, 116
integral —closure 397
—domain 116
—element 395
—ring extension 395
integrally closed ring 397
intermediate field 231
closed— 247
stable— 250
intermediate value theorem
167
internal —direct product 131
—direct sum 62, 175
—weak direct product 62
Interpolation Lagrange's—Formula, 166
intersection 3
invariant —dimension property 185
—factors 80, 225, 357, 361
—subspace 356
inverse 24, 116
—image 4, 31
—of a matrix 331
two sided—of a function 5
invertible —element 116
—fractional ideal 401
—matrix 331
irreducible —element 136
—module 416
—polynomial 161, 234
—variety, 413
irredundant primary de- composition 381
isolated prime ideal 384
isomorphism 30, 118, 170
—theorems 44, 125ff,
172ff natural— 468
isotropy group 89
join of groups 33
Jordan —canonical form 361
elementary—matrix 359
—Holder Theorem 111, 375
Jacobson
—Density Theorem 420
—radical 426fT
—semisimple 429
A^-algebra, see algebra
A>homomorphism 243
Kaplansky, I. vi, 251, 391,
432
kernel, 31, 119, 170
difference— 482
—of a morphism 483
Klein four group 29
Kronecker
—delta 204
—'s method 167
Krull
—Intersection Theorem,
389
— Schmidt Theorem, 86
l.u.b., 14
Lagrange's
—Interpolation Formula
166
—Theorem 39
lattice 14
complete— 15
Law of Well Ordering 10
leading coefficient 150
least
—element 13
—upper bound 14
left
—adjoint 477ff
—annihilator 444
—coset 38
—exact 201
—Goldie ring 447ff
—ideal 122
—inverse of a function 5
—invertible element 116
—order 447
—quasi-regular 426
—quotient ring 447
length of a series 107, 375
Levitsky's Theorem 434
linear
—algebra 327ff
—combination 71
—dependence 181
—equations 346
—functional 203
—independence 74, 181,
291
—ordering 13
INDEX
linear transformation 170
characteristic polynomi-
al of a— 366ff
decomposition of a—
355ff
eigenvalues of a— 367
eigenvectors of a— 367
elementary divisors of a
— 357
invariant factors of a—
357
matrix of a— 329, 333
nullity of a— 335
rank of a— 335ff
linearly disjoint subfields
318ff
linearly independent
—automorphisms, 291
—elements 74, 181
local ring 147
localization 147
logic 1
lower bound 14
Lying-over Theorem 398
w-system 449
Mac Lane, S. xiii, 325,464
main diagonal 328
map, see function
Maschke's Theorem 454
mathematical induction 10
matrices, 328ff
characteristic polynomi-
als of— 366ff
companion— 358
determinants of— 351 ff
direct sum of— 360
eigenvalues of— 368
eigenvectors of— 368
elementary— 338
elementary divisors of
— 361
elementary Jordan— 359
equivalent— 332, 335ff
—in row echelon form
346
invariant factors of—
361
invertible— 331
minors of— 352
—of a homomorphism
329ff, 333
rank of— 336, 337
row spaces of— 336
secondary— 340
similar— 332, 355ff
skew-symmetric— 335
symmetric— 335
trace of— 369
triangular— 335
maximal
—element 13
—ideal 127ff
—normal subgroup 108
—subfield 457
maximum condition 373
McBrien, V.O. xiii, 121
McCoy radical 444ff
McKay, J.H. 93
meaningful product 27
membership 2
metacyclic group 99
middle linear map 207, 217
canonical— 209
minimal
—annihilator 226
—element 373
—left ideal 416
—normal subgroup 103
—polynomial 234, 356
—prime ideal 382
minimum
—condition 373
—element 13
—polynomial 234
minor of a matrix 352
modular left ideal 417
module(s) 168ff, 371
algebra— 451
Artinian— 372ff
completely reducible—
437
direct sum of— 173, 175
direct product of— 173
dual— 2031T
exact sequence of— 175ff
faithful—418
free— 181ff, 188
homomorphism of—
170ff, 199ff
injective— 193ff
isomorphism theorems
for— 172ff
Noetherian—372ff
—over a principal ideal
domain 218ff
projective— 190ff
rank of— 185
reflexive— 205
semisimple— 437
simple— 416
suçn of— 171
tensor product of—
208ff
torsion— 179, 220
torsion-free— 220
trivial— 170
unitary— 169
Moebius function 301
monic
—morphism 480
—polynomial 150
Monk, G.S. ix, 222
monoid 24
monomial 152
degree of a— 157
monomorphism 30, 118,
170
—in a category 480
morphism 52, 480ff
cokernel of a— 483
epic— 480
kernel of a— 483
monic— 480
zero— 482
multilinear
—form 349fF
—function 349ff
multiple root 161, 261
multiplicative set 142
multiplicity of a root 161
mutatis mutandis xv
/?-th root of unity, see unity
rt-fold transitive 424
Nakayama's Lemma 389
natural
—homomorphism 469
—isomorphism 468
—numbers 9
—transformation 468fT
nil
—ideal 430
—radical 379, 450
nilpotent
—element 121, 430
—endomorphism 85
—group lOOff
—ideal 430
Noether
—Normalization
Lemma 410
—Skolem Theorem 460
Noetherian
—modules 372ff, 387ff
—rings 372, 387ff
INDEX
nonsingular matrix 331
norm 289
normal
—closure 265
—endomorphism 84
—extension field 264ff
—series 107ff, 375
—subgroup 4Iff
normalization lemma 410
normalizer 89
null set 3
nullity 335
Nullstellensatz 412
number
cardinal— 16ff
natural— 9
Nunke, R. J. x, xiii, 93
object
—in a category 52
initial— 57
terminal— 57
odd permutation 48
one-to-one
—correspondence 5
—function 4
onto function 5
operation
binary— 24
elementary row or
column— 338
opposite
—category 466
—ring, 122, 330
orbit 89
—of a permutation 47
order
—ideal 220
left— 447
—of an element 35, 220
—of a group 24
partial— 13
ordered pair 6
ordering
—by extension 18
linear— 13
partial— 13
simple— 13
total— 13
orthogonal idempotents
135, 439
P.I.D., see principal ideal
domain
p-group 93
P-primary 380, 384
P-radical 425
pair
—formation 6
ordered— 6
partial ordering 13
partition 7
perfect field 289
permutation(s) 26, 46ff
disjoint— 47
even— 48
odd— 48
orbits of— 47
sign of a— 48
Polynomial(s)
characteristic— 366ff
coefficients of— 150
content of a— 162
cyclotomic— 166, 298
degree of— 157, 158
discriminant of— 270
Galois group of a—
269ff
general—of degree n 308
—in n indeterminates
151
irreducible— 161, 234
minimal— 234, 356
minimum— 234
monic— 150
primitive— 162
ring of— 149ff, 151, 156
roots of— 160
separable— 261
power
—axiom 3
—series 154fT
—set 3
predecessor
immediate— 14
presentation of a group 67
primary
—decomposition, 381,
383ff
—ideal 380ff
P-primary ideal 380
P-primary submodule
384
—rational canonical
form 361
—submodule 383ff
prime
associated—ideal 380,
387
—element 136
embedded—ideal 384
—ideal 126ff, 377ff —integer 11
isolated—ideal 384
minimal—ideal 382
—radical 379, 444ff relatively—integers 11
relatively—elements 140
—ring 445flf subfield 279
primitive —element theorem 287, 288
—ideal 425
—polynomial 162
—ring 418ff —root of unity 295
principal —ideal 123
—ideal ring 123
—ideal domain 123
modules over a—ideal
domain 218fF
principle —of mathematical
induction 10
—of transfinite induc- tion 14
—of well ordering 14
product —category 467
Cartesian— 6, 7ff direct— 59ff, 130, 131, 173
—in a category 53fT
—map 228
semidirect— 99
subdirect— 434
weak direct— 60, 62
projection canonical— 8, 59, 130, 173
projective module 190ff proper —class 2
—ideal 123
—refinement 108, 375
—subgroup 32
—values 367
—vectors 367
Prüfer domain 409
pullback 170, 484
purely inseparable —element 282
—extension 282ff
purely transcendental ex- tension 314
quasi-inverse, 426
quasi-regular 426
quaternion group 33,97,99
quaternions
division ring of real—
117,461
quotient —class 6
—field 144
—group 42ff —ring 125, 144ff, 447
r-cycle 46
/^-module, see module
Rad 1379
radical —extension field 303ff Jacobson— 426ff nil— 379, 450
—of an ideal 379
P-radical 425
prime— 379, 444ff —property 425
—ring 429
solvable by—s 303
range of a function 3
rank
column— 336, 339
—of a free abelian group 72
—of a free module 185
—of a homomorphism
335ff, 339
row— 336, 339
—of a matrix 337ff rational —canonical form 360
—function 233
symmetric—function 252ff
rationals modulo one 27
Recursion Theorem 10
reduced —abelian group 198
—primary decomposi- tion 381, 384
—row echelon form 346
—word 64, 68
refinement of a series 108,
375
reflexive —module 205
—relation 6
regular —element 447
—function 412
—left ideal 417
—left quasi—, 426
Von Neumann—ring 442
relative complement 3
relation 6, 66
antisymmetric— 13
congruence— 27
equivalence— 6
generators and—s, 67ff,
343AF
reflexive— 6
symmetric— 6
transitive— 6
relative —dimension 245
—index 245
relatively prime —integers 11
—ring elements 140
Remainder Theorem 159
Chinese— 131
representable functor 470
representation 470
résolvant cubic 272
restriction 4
right —adjoint 477
—annihilator 444
—coset 38
—Goldie ring 447
—ideal 122
—inverse of a function 5
—invertible element 116
-quasi-regular 426
—quotient ring 447
ring(s) 115ff, 371 fF, 414ff Artinian— 372, 421,435
Boolean— 120
commutative— 115, 37 Iff
direct product of— 130, 131
discrete valuation— 401
division— 116, 462
endomorphism— 415
Euclidean— 139
—extensions 394ff Goldmann— 400
group— 117
homomorphism of—
118
integrally closed— 397
left quotient— 447
local— 147
Noetherian— 372, 387ff —of formal power series 154ff
—of polynomials 149ff —of quotients or frac- tions 144ff opposite— 122, 330
primitive— 418ff prime— 445ff quotient— 125, 447
radical— 429
regular— 442
semi prime— 444ff
semisimple— 429, 434ff
simple— 416ff
subdirectly irreducible—
442
root
adjoining a— 236
multiple— 161, 261
multiplicity of a— 161
—of unity 294
simple— 161, 261
row
—echelon form 346
elementary—operation 338
—rank 336, 339
—space 336
—vector 329
ruler and compass con- structions 238
Russell's Paradox 2
scalar matrix 328
Schreier's Theorem 110, 375
Schroeder-Bernstein
Theorem 17
Schur's Lemma 419
second isomorphism
theorem 44, 126, 173
secondary matrix 340
semidirect product 99
semigroup 24
semiprime ring 444ff semisimple —module 437
—ring 429, 434ff separable —degree 285ff —element 261
—extension 261, 282ff, 324
—polynomial 261
separably generated exten- sion 322
separating transcendence
base 322
sequence 11
exact— 175ff short exact— 176
series
ascending central— 100
composition— 108, 375
equivalent— 109, 375
formal power— 154ff normal— 107ff, 375
refinement of a— 108, 375
solvable— 108
subnormal— 107ff set(s) denumerable— 16
disjoint— 3
empty— 3
equipollent— 15
finite— 16
infinite— 16
linearly ordered— 13
multiplicative— 142
null— 3
partially ordered— 13
power— 3
underlying— 55
well ordered— 13
Short —Five Lemma 176
—exact sequence 176
sign of a permutation 48
similar matrices 332, 355ff simple —components 440
—extension field 232
—group 49
—module 179, 375,416ff —ordering 13
—ring 416ff —root, 161, 261
singleton 6
skew-symmetric —matrix 335
—multilinear form 349
solvable —by radicals 303
—group 102ff, 108
—series 108
span 181
spectrum 378
split exact sequence 177
splitting fields 257ff stabilizer 89
stable intermediate field
250
standard
—basis of Rn 336
—/i-product 28
subalgebra 228
subclass 2
subdirect product 434
subdirectly irreducible ring
442
subfield(s) composite— 233
—generated by a set 231, 232
linearly disjoint— 318ff maximal— 457
prime— 279
subgroup(s) 3Iff characteristic— 103
closed— 247
commutator— 102
cyclic— 32
derived— 102
fully invariant— 103
—generated by a set 32
—generated by groups 33
join of— 33
maximal normal— 108
minimal normal— 103
normal— 4 Iff proper— 32
Sylow— 94
transitive— 92, 269
trivial— 32
submodule(s) 171
chain conditions on—
372ff cyclic— 171
finitely generated—171
—generated (or spanned)
by a set 171
primary— 383ff sum of— 171
torsion— 220
subnormal series 107ff subring 122
—generated by a set 231, 232, 395
subset 3
subspace 171
</>-invariant— 356
substitution homomor-
phism 153
successor
immediate— 15
sum (= coproduct) 54ff sum
direct— 60, 62, 173, 175
—of submodules 171
summand
direct— 63, 437
surjective function 5
Swords, R.J., xiii Sylow —p-subgroup 94
—theorems 92ff symmetric —group 26, 46ff —matrix 335
—multilinear function 349
—rational function 252ff —relation 6
symmetries of the square 26
tensor product 208ff —of algebras 229
induced homomorphism on the— 209
terminal object 57
third isomorphism theorem
44, 126, 173
torsion —group 78
—module 179, 220
—subgroup 78
—submodule 179, 220
torsion-free —group 78
—module 220
total —degree 157
—ordering 13
trace 289, 369
transcendence
—base 313fF
—degree 316
separating—base 322
transcendental —element 233
—extension 233, 31 Iff purely—extension 314
transfinite induction 14
transformation linear— 170, 355ff natural— 468ff transitive —relation 6
—subgroup 92, 269
—subring 424
translation 88
transpose of a matrix 328
transposition 46
triangular matrix 335
trichotomy law 18
trivial ideal 123
U.F.D., see unique factori- zation domain underlying set 55
union 3
disjoint— 58
unique factorization do- main 137
unit 116
—map 228
unitary module 169
unity root of— 294
primitive root of— 295
universal —element 470
—mapping property 9, 57
—object 57
upper bound 13
valuation —domain 409
discrete—ring 404
Van Dyck's Theorem 67
variety 409
vector column— 333
row— 329
vector space 169, 180ff
finite dimensional— 186
Von Neumann regular ring 442
weak direct product 60, 62
Wedderburn's Theorem on finite division rings 462
Wedderburn-Artin
Theorems 421,435
well ordered set 13
well ordering law of— 10
—principle 14
word 64, 68
reduced— 64, 68
empty— 64
Yoneda, N. 472
Zassenhaus Lemma, 109,
375
zero 409
—divisor 116
—element 115
—matrix 328
—morphism 482
—object 481
—of a polynomial 160
Zorn's Lemma, 13
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