Algebra / P. M. Cohn.
Editor: Chichester [Sussex, England] : Wiley, c1982-<c1991Edición: 2nd edDescripción: v. <1-3 > : il. ; 23-24 cmISBN: 0471101699 (v. 1 : pbk); 0471101680 (v. 1); 047192234X (v. 2); 0471922358 (v. 2 : pbk); 0471928402 (v. 3)Tema(s): AlgebraOtra clasificación: 00A05 | 00A05 (08-01 13-01 20-01)Preface to Second Edition vii From the Preface to First Edition ix Table of interdependence of chapters xv 1 Sets and mappings 1.1 The need for logic [1] 1.2 Sets [8] 1.3 Mappings [12] 1.4 Equivalence relations [17] 1.5 Ordered sets [19] Further exercises [21] 2 Integers and rational numbers 2.1 The integers [22] 2.2 Divisibility and factorization in Z [26] 2.3 Congruences [30] 2.4 The rational numbers and some finite fields [37] Further exercises [39] 3 Groups 3.1 Monoids [42] 3.2 Groups; the axioms [45] 3.3 Group actions and coset decompositions [51] 3.4 Cyclic groups [56] 3.5 Permutation groups [58] 3.6 Symmetry [53] Further exercises [67] 4 Vector spaces and linear mappings 4.1 Vectors and linear dependence [70] 4.2 Linear mappings [74] 4.3 Bases and dimension [76] 4.4 Direct sums and quotient spaces [81] 4.5 The space of linear mappings [86] 4.6 Change of basis [93] 4.7 The rank [96] 4.8 Affine spaces [101] 4.9 Category and functor [104] Further exercises [110] 5 Linear equations 5.1 Systems of linear equations [112] 5.2 Elementary operations [115] 5.3 Linear programming [120] 5.4 PAQ-reduction and the inversion of matrices [129] 5.5 Block multiplication [132] Further exercises [134] 6 Rings and fields 6.1 Definition and examples [136] 6.2 The field of fractions of an integral domain [140] 6.3 The characteristic [145] 6.4 Polynomials [147] 6.5 Factorization [153] 6.6 The zeros of polynomials [161] 6.7 The factorization of polynomials [164] 6.8 Derivatives [168] 6.9 Symmetric and alternating functions [177] Further exercises [182] 7 Determinants 7.1 Definition and basic properties [186] 7.2 Expansion of a determinant [194] 7.3 The determinantal rank [199] 7.4 The resultant [201] Further exercises [205] 8 Quadratic forms 8.1 Bilinear forms and pairings [208] 8.2 Dual spaces [211] 8.3 Inner products; quadratic and hermitian forms [216] 8.4 Euclidean and unitary spaces [224] 8.5 Orthogonal and unitary matrices [230] 8.6 Alternating forms [241] Further exercises [245] 9 Further group theory 9.1 The isomorphism theorems [249] 9.2 The Jordan-Holder theorem [255] 9.3 Groups with operators [259] 9.4 Automorphisms [263] 9.5 The derived group; soluble groups and simple groups [266] 9.6 Direct products [272] 9.7 Abelian groups [279] 9.8 The Sylow theorems [287] 9.9 Generators and defining relations; free groups [292] Further exercises [298] 10 Rings and modules 10.1 . Ideals and quotient rings [301] 10.2 Modules over a ring [304] 10.3 Direct products and direct sums [311] 10.4 Free modules [314] 10.5 Principal ideal domains [318] 10.6 Modules over a principal ideal domain [326] Further exercises [332] 11 Normal forms for matrices 11.1 Eigenvalues and eigenvectors [335] 11.2 The k[x]-module defined by an endomorphism [339] 11.3 Cyclic endomorphisms [344] 11.4 The Jordan normal form [347] 11.5 The Jordan normal form: another method [352] 11.6 Normal matrices [356] 11.7 Linear algebras [358] Further exercises [360] Solutions to the exercises [362] Appendices 1 Further reading [400] 2 Some frequently used notations [401] Index [403]
Contents Preface to the Second Edition ix From the Preface to the First Edition xi Conventions on terminology xiii Table of interdependence of chapters (Leitfaden) xv 1 Sets 1.1 Finite, countable and uncountable sets [1] 1.2 Zorn’s lemma and well-ordered sets [8] 1.3 Categories [16] 1.4 Graphs [21] Further exercises [27] 2 Lattices 2.1 Definitions; modular and distributive lattices [30] 2.2 Chain conditions [38] 2.3 Boolean algebras [45] 2.4 Möbius functions [54] Further exercises [58] 3 Field theory 3.1 Fields and their extensions [62] 3.2 Splitting fields [69] 3.3 The algebraic closure of a field [74] 3.4 Separability [77] 3.5 Automorphisms of field extensions [80] 3.6 The fundamental theorem of Galois theory [85] 3.7 Roots of unity [91] 3.8 Finite fields [97] 3.9 Primitive elements; norm and trace [102] 3.10 Galois theory of equations [107] 3.11 The solution of equations by radicals [113] Further exercises [121] 4 Modules 4.1 The category of modules over a ring [124] 4.2 Semisimple modules [130] 4.3 Matrix rings [135] 4.4 Free modules [140] 4.5 Projective and injective modules [146] 4.6 Duality of finite abelian groups [152] 4.7 The tensor product of modules [155] Further exercises [163] 5 Rings and algebras 5.1 Algebras: definition and examples [165] 5.2 Direct products of rings [170] 5.3 The Wedderbum structure theorems [174] 5.4 The radical [178] 5.5 The tensor product of algebras [183] 5.6 The regular representation; norm and trace [187] 5.7 Composites of fields [191] Further exercises [195] 6 Quadratic forms and ordered fields 6.1 Inner product spaces [197] 6.2 Orthogonal sums and diagonalization [200] 6.3 The orthogonal group of a space [204] 6.4 Witt’s cancellation theorem and the Witt group of a field [208] 6.5 Ordered fields [212] 6.6 The field of real numbers [215] Further exercises [220] 7 Representation theory of finite groups 7.1 Basic definitions [221] 7.2 The averaging lemma and Maschke’s theorem [226] 7.3 Orthogonality and completeness [229] 7.4 Characters [233] 7.5 Complex representations [241] 7.6 Representations of the symmetric group [247] 7.7 Induced representations [253] 7.8 Applications: the theorems of Burnside and Frobenius [258] Further exercises [262] 8 Valuation theory 8.1 Divisibility and valuations [264] 8.2 Absolute values [269] 8.3 The p-adic numbers [280] 8.4 Integral elements [289] 8.5 Extension of valuations [294] Further exercises [302] 9 Commutative rings 9.1 Operations on ideals [305] 9.2 Prime ideals and factorization [307] 9.3 Localization [310] 9.4 Noetherian rings [317] 9.5 Dedekind domains [319] 9.6 Modules over Dedekind domains [329] 9.7 Algebraic equations [334] 9.8 The primary decomposition [338] 9.9 Dimension [345] 9.10 The Hilbert Nullstellensatz [350] Further exercises [353] 10 Coding theory 10.1 The transmission of information [356] 10.2 Block codes [358] 10.3 Linear codes [361] 10.4 Cyclic codes [369] 10.5 Other codes [374] Further exercises [378] 11 Languages and automata 11.1 Monoids and monoid actions [379] 11.2 Languages and grammars [383] 11.3 Automata [387] 11.4 Variable-length codes [395] 11.5 Free algebras and formal power series rings [404] Further exercises [413] Bibliography [415] List of notations [418] Index [421]
Contents Preface to the Second Edition ix Conventions on terminology and notes to the reader xi 1 Universal algebra 1.1 Algebras and homomorphisms [1] 1.2 Congruences and the isomorphism theorems [4] 1.3 Free algebras and varieties [11] 1.4 Abstract dependence relations [20] 1.5 The diamond lemma [25] 1.6 Ultraproducts [27] 1.7 The natural numbers [31] Further exercises [38] 2 Multilinear algebra 2.1 Graded algebras [40] 2.2 Free algebras and tensor algebras [43] 2.3 The Hilbert series of a graded algebra or module [52] 2.4 The exterior algebra on a module [58] Further exercises [66] 3 Homological algebra 3.1 Additive and abelian categories [69] 3.2 Functors on abelian categories [78] 3.3 The category MR [87] 3.4 Homological dimension [94] 3.5 Derived functors [99] 3.6 Ext and Tor and the global dimension of a ring [110] Further exercises [120] 4 Further group theory 4.1 Group extensions [124] 4.2 The Frattini subgroup and the Fitting subgroup [135] 4.3 Hall subgroups [141] 4.4 The transfer [146] 4.5 Free groups [148] 4.6 Commutators [154] 4.7 Linear groups [159] Further exercises [165] 5 Further field theory 5.1 Algebraic dependence [168] 5.2 Simple transcendental extensions [171] 5.3 Separable and p-radical extensions [176] 5.4 Derivations [181] 5.5 Linearly disjoint extensions [186] 5.6 Infinite algebraic extensions [196] 5.7 Galois cohomology [202] 5.8 Kummer extensions [206] Further exercises [210] 6 Algebras 6.1 The Krull-Schmidt theorem [213] 6.2 The projective cover of a module [218] 6.3 Semiperfect rings [221] 6.4 Equivalence of module categories [227] 6.5 The Morita context [234] 6.6 Projective, injective and flat modules [239] 6.7 Hochschild cohomology and separable algebras [248] Further exercises [256] 7 Central simple algebras 7.1 Simple Artinian rings [258] 7.2 The Brauer group [266] 7.3 The reduced norm and trace [273] 7.4 Quaternion algebras [280] 7.5 Crossed products [283] 7.6 Change of base field [289] 7.7 Cyclic algebras [295] Further exercises [298] Ouadratic forms and ordered fields 8 1 The Clifford algebra of a quadratic space [300] 8.2 The spinor norm [306] 8.3 Formally real fields [309] 8 4 The Witt ring of a field [322] 8.5 The symplectic group [330] 8.6 The orthogonal group [337] 8.7 Quadratic forms in characteristic 2 [344] Further exercises [347] 9 Noetherian rings and polynomial identities 9.1 Rings of fractions [349] 9.2 Principal ideal domains [355] 9.3 Skew polynomials and Laurent series [359] 9.4 Goldie’s theorem [367] 9.5 PI-algebras [375] 9.6 Varieties of PI-algebras and Regev’s theorem [380] 9.7 Generic matrix rings and central polynomials [384] 9.8 Generalized polynomial identities [389] Further exercises [393] 10 Rings without finiteness assumptions 10.1 The endomorphism ring of a vector space [395] 10.2 The density theorem revisited [398] 10.3 Primitive rings [401] 10.4 Semiprimitive rings and the Jacobson radical [404] 10.5 Algebras without a unit element [409] 10.6 Semiprime rings and nilradicals [413] 10.7 Prime PI-algebras [419] 10.8 Simple algebras with minimal right ideals [423] 10.9 Firs and semifirs [425] Further exercises [430] 11 Skew fields 11.1 Generalities [432] 11.2 The Dieudonne determinant [436] 11.3 Free fields [443] 11.4 Valuations on skew fields [448] 11.5 Pseudo-linear extensions [455] Further exercises [459] Bibliography [461] List of notations [464] Index [468]
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Incluye referencias bibliográficas e índices.
Preface to Second Edition vii --
From the Preface to First Edition ix --
Table of interdependence of chapters xv --
1 Sets and mappings --
1.1 The need for logic [1] --
1.2 Sets [8] --
1.3 Mappings [12] --
1.4 Equivalence relations [17] --
1.5 Ordered sets [19] --
Further exercises [21] --
2 Integers and rational numbers --
2.1 The integers [22] --
2.2 Divisibility and factorization in Z [26] --
2.3 Congruences [30] --
2.4 The rational numbers and some finite fields [37] --
Further exercises [39] --
3 Groups --
3.1 Monoids [42] --
3.2 Groups; the axioms [45] --
3.3 Group actions and coset decompositions [51] --
3.4 Cyclic groups [56] --
3.5 Permutation groups [58] --
3.6 Symmetry [53] --
Further exercises [67] --
4 Vector spaces and linear mappings --
4.1 Vectors and linear dependence [70] --
4.2 Linear mappings [74] --
4.3 Bases and dimension [76] --
4.4 Direct sums and quotient spaces [81] --
4.5 The space of linear mappings [86] --
4.6 Change of basis [93] --
4.7 The rank [96] --
4.8 Affine spaces [101] --
4.9 Category and functor [104] --
Further exercises [110] --
5 Linear equations --
5.1 Systems of linear equations [112] --
5.2 Elementary operations [115] --
5.3 Linear programming [120] --
5.4 PAQ-reduction and the inversion of matrices [129] --
5.5 Block multiplication [132] --
Further exercises [134] --
6 Rings and fields --
6.1 Definition and examples [136] --
6.2 The field of fractions of an integral domain [140] --
6.3 The characteristic [145] --
6.4 Polynomials [147] --
6.5 Factorization [153] --
6.6 The zeros of polynomials [161] --
6.7 The factorization of polynomials [164] --
6.8 Derivatives [168] --
6.9 Symmetric and alternating functions [177] --
Further exercises [182] --
7 Determinants --
7.1 Definition and basic properties [186] --
7.2 Expansion of a determinant [194] --
7.3 The determinantal rank [199] --
7.4 The resultant [201] --
Further exercises [205] --
8 Quadratic forms --
8.1 Bilinear forms and pairings [208] --
8.2 Dual spaces [211] --
8.3 Inner products; quadratic and hermitian forms [216] --
8.4 Euclidean and unitary spaces [224] --
8.5 Orthogonal and unitary matrices [230] --
8.6 Alternating forms [241] --
Further exercises [245] --
9 Further group theory --
9.1 The isomorphism theorems [249] --
9.2 The Jordan-Holder theorem [255] --
9.3 Groups with operators [259] --
9.4 Automorphisms [263] --
9.5 The derived group; soluble groups and simple groups [266] --
9.6 Direct products [272] --
9.7 Abelian groups [279] --
9.8 The Sylow theorems [287] --
9.9 Generators and defining relations; free groups [292] --
Further exercises [298] --
10 Rings and modules --
10.1 . Ideals and quotient rings [301] --
10.2 Modules over a ring [304] --
10.3 Direct products and direct sums [311] --
10.4 Free modules [314] --
10.5 Principal ideal domains [318] --
10.6 Modules over a principal ideal domain [326] --
Further exercises [332] --
11 Normal forms for matrices --
11.1 Eigenvalues and eigenvectors [335] --
11.2 The k[x]-module defined by an endomorphism [339] --
11.3 Cyclic endomorphisms [344] --
11.4 The Jordan normal form [347] --
11.5 The Jordan normal form: another method [352] --
11.6 Normal matrices [356] --
11.7 Linear algebras [358] --
Further exercises [360] --
Solutions to the exercises [362] --
Appendices --
1 Further reading [400] --
2 Some frequently used notations [401] --
Index [403] --
Contents --
Preface to the Second Edition ix --
From the Preface to the First Edition xi --
Conventions on terminology xiii --
Table of interdependence of chapters (Leitfaden) xv --
1 Sets --
1.1 Finite, countable and uncountable sets [1] --
1.2 Zorn’s lemma and well-ordered sets [8] --
1.3 Categories [16] --
1.4 Graphs [21] --
Further exercises [27] --
2 Lattices --
2.1 Definitions; modular and distributive lattices [30] --
2.2 Chain conditions [38] --
2.3 Boolean algebras [45] --
2.4 Möbius functions [54] --
Further exercises [58] --
3 Field theory --
3.1 Fields and their extensions [62] --
3.2 Splitting fields [69] --
3.3 The algebraic closure of a field [74] --
3.4 Separability [77] --
3.5 Automorphisms of field extensions [80] --
3.6 The fundamental theorem of Galois theory [85] --
3.7 Roots of unity [91] --
3.8 Finite fields [97] --
3.9 Primitive elements; norm and trace [102] --
3.10 Galois theory of equations [107] --
3.11 The solution of equations by radicals [113] --
Further exercises [121] --
4 Modules --
4.1 The category of modules over a ring [124] --
4.2 Semisimple modules [130] --
4.3 Matrix rings [135] --
4.4 Free modules [140] --
4.5 Projective and injective modules [146] --
4.6 Duality of finite abelian groups [152] --
4.7 The tensor product of modules [155] --
Further exercises [163] --
5 Rings and algebras --
5.1 Algebras: definition and examples [165] --
5.2 Direct products of rings [170] --
5.3 The Wedderbum structure theorems [174] --
5.4 The radical [178] --
5.5 The tensor product of algebras [183] --
5.6 The regular representation; norm and trace [187] --
5.7 Composites of fields [191] --
Further exercises [195] --
6 Quadratic forms and ordered fields --
6.1 Inner product spaces [197] --
6.2 Orthogonal sums and diagonalization [200] --
6.3 The orthogonal group of a space [204] --
6.4 Witt’s cancellation theorem and the Witt group of a field [208] --
6.5 Ordered fields [212] --
6.6 The field of real numbers [215] --
Further exercises [220] --
7 Representation theory of finite groups --
7.1 Basic definitions [221] --
7.2 The averaging lemma and Maschke’s theorem [226] --
7.3 Orthogonality and completeness [229] --
7.4 Characters [233] --
7.5 Complex representations [241] --
7.6 Representations of the symmetric group [247] --
7.7 Induced representations [253] --
7.8 Applications: the theorems of Burnside and Frobenius [258] --
Further exercises [262] --
8 Valuation theory --
8.1 Divisibility and valuations [264] --
8.2 Absolute values [269] --
8.3 The p-adic numbers [280] --
8.4 Integral elements [289] --
8.5 Extension of valuations [294] --
Further exercises [302] --
9 Commutative rings --
9.1 Operations on ideals [305] --
9.2 Prime ideals and factorization [307] --
9.3 Localization [310] --
9.4 Noetherian rings [317] --
9.5 Dedekind domains [319] --
9.6 Modules over Dedekind domains [329] --
9.7 Algebraic equations [334] --
9.8 The primary decomposition [338] --
9.9 Dimension [345] --
9.10 The Hilbert Nullstellensatz [350] --
Further exercises [353] --
10 Coding theory --
10.1 The transmission of information [356] --
10.2 Block codes [358] --
10.3 Linear codes [361] --
10.4 Cyclic codes [369] --
10.5 Other codes [374] --
Further exercises [378] --
11 Languages and automata --
11.1 Monoids and monoid actions [379] --
11.2 Languages and grammars [383] --
11.3 Automata [387] --
11.4 Variable-length codes [395] --
11.5 Free algebras and formal power series rings [404] --
Further exercises [413] --
Bibliography [415] --
List of notations [418] --
Index [421] --
Contents --
Preface to the Second Edition ix --
Conventions on terminology and notes to the reader xi --
1 Universal algebra --
1.1 Algebras and homomorphisms [1] --
1.2 Congruences and the isomorphism theorems [4] --
1.3 Free algebras and varieties [11] --
1.4 Abstract dependence relations [20] --
1.5 The diamond lemma [25] --
1.6 Ultraproducts [27] --
1.7 The natural numbers [31] --
Further exercises [38] --
2 Multilinear algebra --
2.1 Graded algebras [40] --
2.2 Free algebras and tensor algebras [43] --
2.3 The Hilbert series of a graded algebra or module [52] --
2.4 The exterior algebra on a module [58] --
Further exercises [66] --
3 Homological algebra --
3.1 Additive and abelian categories [69] --
3.2 Functors on abelian categories [78] --
3.3 The category MR [87] --
3.4 Homological dimension [94] --
3.5 Derived functors [99] --
3.6 Ext and Tor and the global dimension of a ring [110] --
Further exercises [120] --
4 Further group theory --
4.1 Group extensions [124] --
4.2 The Frattini subgroup and the Fitting subgroup [135] --
4.3 Hall subgroups [141] --
4.4 The transfer [146] --
4.5 Free groups [148] --
4.6 Commutators [154] --
4.7 Linear groups [159] --
Further exercises [165] --
5 Further field theory --
5.1 Algebraic dependence [168] --
5.2 Simple transcendental extensions [171] --
5.3 Separable and p-radical extensions [176] --
5.4 Derivations [181] --
5.5 Linearly disjoint extensions [186] --
5.6 Infinite algebraic extensions [196] --
5.7 Galois cohomology [202] --
5.8 Kummer extensions [206] --
Further exercises [210] --
6 Algebras --
6.1 The Krull-Schmidt theorem [213] --
6.2 The projective cover of a module [218] --
6.3 Semiperfect rings [221] --
6.4 Equivalence of module categories [227] --
6.5 The Morita context [234] --
6.6 Projective, injective and flat modules [239] --
6.7 Hochschild cohomology and separable algebras [248] --
Further exercises [256] --
7 Central simple algebras --
7.1 Simple Artinian rings [258] --
7.2 The Brauer group [266] --
7.3 The reduced norm and trace [273] --
7.4 Quaternion algebras [280] --
7.5 Crossed products [283] --
7.6 Change of base field [289] --
7.7 Cyclic algebras [295] --
Further exercises [298] --
Ouadratic forms and ordered fields --
8 1 The Clifford algebra of a quadratic space [300] --
8.2 The spinor norm [306] --
8.3 Formally real fields [309] --
8 4 The Witt ring of a field [322] --
8.5 The symplectic group [330] --
8.6 The orthogonal group [337] --
8.7 Quadratic forms in characteristic 2 [344] --
Further exercises [347] --
9 Noetherian rings and polynomial identities --
9.1 Rings of fractions [349] --
9.2 Principal ideal domains [355] --
9.3 Skew polynomials and Laurent series [359] --
9.4 Goldie’s theorem [367] --
9.5 PI-algebras [375] --
9.6 Varieties of PI-algebras and Regev’s theorem [380] --
9.7 Generic matrix rings and central polynomials [384] --
9.8 Generalized polynomial identities [389] --
Further exercises [393] --
10 Rings without finiteness assumptions --
10.1 The endomorphism ring of a vector space [395] --
10.2 The density theorem revisited [398] --
10.3 Primitive rings [401] --
10.4 Semiprimitive rings and the Jacobson radical [404] --
10.5 Algebras without a unit element [409] --
10.6 Semiprime rings and nilradicals [413] --
10.7 Prime PI-algebras [419] --
10.8 Simple algebras with minimal right ideals [423] --
10.9 Firs and semifirs [425] --
Further exercises [430] --
11 Skew fields --
11.1 Generalities [432] --
11.2 The Dieudonne determinant [436] --
11.3 Free fields [443] --
11.4 Valuations on skew fields [448] --
11.5 Pseudo-linear extensions [455] --
Further exercises [459] --
Bibliography [461] --
List of notations [464] --
Index [468] --
MR, 83e:00002 (v. 1)
MR, 91b:00001 (v. 2)
MR, 92c:00001 (v. 3)
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