Lectures in abstract algebra. Volume I, Basic concepts / by Nathan Jacobson.
Series The University series in higher mathematicsEditor: Princeton : Van Nostrand, 1951Descripción: xii, 217 p. ; 24 cmOtra clasificación: 08-XXintroduction: concepts from set theory THE SYSTEM OF NATURAL NUMBERS 1. Operations on sets [2] 2. Product sets, mappings [3] 3. Equivalence relations [4] 4. The natural numbers [2] 5. The system of integers [10] 6. The division process in I [12] CHAPTER I: SEMI-GROUPS AND GROUPS 1. Definition and examples of semi-groups [15] 2. Non-associative binary compositions [18] 3. Generalized associative law. Powers [20] 4. Commutativity [21] 5. Identities and inverses [22] 6. Definition and examples of groups [23] 7. Subgroups [24] 8. Isomorphism [26] 9. Transformation groups [27] 10. Realization of a group as a transformation group [28] 11. Cyclic groups. Order of an element [30] 12. Elementary properties of permutations [34] 13. Coset decompositions of a group [37] 14. Invariant subgroups and factor groups [40] 15. Homomorphism of groups [41] 16. The fundamental theorem of homomorphism for groups [43] 17. Endomorphisms, automorphisms, center of a group [45] 18. Conjugate classes [47] CHAPTER II: RINGS, INTEGRAL DOMAINS AND FIELDS 1. Definition and examples [49] 2. Types of rings [53] 3. Quasi-regularity. The circle composition [55] 4. Matrix rings [56] 5. Quaternions [60] 6. Subrings generated by a set of elements. Center [63] 7. Ideals, difference rings [64] 8. Ideals and difference rings for the ring of integers [66] 9. Homomorphism of rings [68] 10. Anti-isomorphism [71] 11. Structure of the additive group of a ring. The charateristic of a ring [74] 12. Algebra of subgroups of the additive group of a ring. Onesided ideals [75] 13. The ring of endomorphisms of a commutative group [78] 14. The multiplications of a ring [82] CHAPTER III: EXTENSIONS OF RINGS AND FIELDS 1. Imbedding of a ring in a ring with an identity [84] 2. Field of fractions of a commutative integral domain [87] 3. Uniqueness of the field of fractions [91] 4. Polynomial rings [92] 5. Structure of polynomial rings [96] 6. Properties of the ring U[x] [97] 7. Simple extensions of a field [100] 8. Structure of any field [103] 9. The number of roots of a polynomial in a field [104] 10. Polynomials in several elements [105] 11. Symmetric polynomials [107] 12. Rings of functions [110] CHAPTER IV: ELEMENTARY FACTORIZATION THEORY 1. Factors, associates, irreducible elements [114] 2. Gaussian semi-groups [115] 3. Greatest common divisors [118] 4. Principal ideal domains [121] 5. Euclidean domains [122] 6. Polynomial extensions of Gaussian domains [124] CHAPTER v: GROUPS WITH OPERATORS 1. Definition and examples of groups with operators [128] 2. M-subgroups, M-factor groups and M-homomorphisms [130] 3. The fundamental theorem of homomorphism for M-groups [132] 4. The correspondence between M-subgroups determined by a homomorphism [133] 5. The isomorphism theorems for M-groups [135] 6. Schreier’s theorem [137] 7. Simple groups and the Jordan-Holder theorem [139] 8. The chain conditions [142] 9. Direct products [144] 10. Direct products of subgroups [145] 11. Projections [149] 12. Decomposition into indecomposable groups [152] 13. The Krull-Schmidt theorem [154] 14. Infinite direct products [159] CHAPTER VI: MODULES AND IDEALS 1. Definitions [162] 2. Fundamental concepts [164] 3. Generators. Unitary modules [166] 4. The chain conditions [168] 5. The Hilbert basis theorem [170] 6. Noetherian rings. Prime and primary ideals [172] 7. Representation of an ideal as intersection of primary ideals [175] 8. Uniqueness theorems [177] 9. Integral dependence [181] 10. Integers of quadratic fields [184] CHAPTER VII: LATTICES 1. Partially ordered sets [187] 2. Lattices [189] 3. Modular lattices [193] 4. Schreier’s theorem. The chain conditions [197] 5. Decomposition theory for lattices with ascending chain condition [201] 6. Independence [202] 7. Complemented modular lattices [205] 8. Boolean algebras [207] Index [213]
| Item type | Home library | Shelving location | Call number | Materials specified | Copy number | Status | Date due | Barcode | Course reserves |
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 08 J17 (Browse shelf) | Vol. I | Available | A-177 | |||
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 08 J17 (Browse shelf) | Vol. I | Ej. 2 | Available | A-2921 |
introduction: concepts from set theory --
THE SYSTEM OF NATURAL NUMBERS --
1. Operations on sets [2] --
2. Product sets, mappings [3] --
3. Equivalence relations [4] --
4. The natural numbers [2] --
5. The system of integers [10] --
6. The division process in I [12] --
CHAPTER I: SEMI-GROUPS AND GROUPS --
1. Definition and examples of semi-groups [15] --
2. Non-associative binary compositions [18] --
3. Generalized associative law. Powers [20] --
4. Commutativity [21] --
5. Identities and inverses [22] --
6. Definition and examples of groups [23] --
7. Subgroups [24] --
8. Isomorphism [26] --
9. Transformation groups [27] --
10. Realization of a group as a transformation group [28] --
11. Cyclic groups. Order of an element [30] --
12. Elementary properties of permutations [34] --
13. Coset decompositions of a group [37] --
14. Invariant subgroups and factor groups [40] --
15. Homomorphism of groups [41] --
16. The fundamental theorem of homomorphism for groups [43] --
17. Endomorphisms, automorphisms, center of a group [45] --
18. Conjugate classes [47] --
CHAPTER II: RINGS, INTEGRAL DOMAINS AND FIELDS --
1. Definition and examples [49] --
2. Types of rings [53] --
3. Quasi-regularity. The circle composition [55] --
4. Matrix rings [56] --
5. Quaternions [60] --
6. Subrings generated by a set of elements. Center [63] --
7. Ideals, difference rings [64] --
8. Ideals and difference rings for the ring of integers [66] --
9. Homomorphism of rings [68] --
10. Anti-isomorphism [71] --
11. Structure of the additive group of a ring. The charateristic of a ring [74] --
12. Algebra of subgroups of the additive group of a ring. Onesided ideals [75] --
13. The ring of endomorphisms of a commutative group [78] --
14. The multiplications of a ring [82] --
CHAPTER III: EXTENSIONS OF RINGS AND FIELDS --
1. Imbedding of a ring in a ring with an identity [84] --
2. Field of fractions of a commutative integral domain [87] --
3. Uniqueness of the field of fractions [91] --
4. Polynomial rings [92] --
5. Structure of polynomial rings [96] --
6. Properties of the ring U[x] [97] --
7. Simple extensions of a field [100] --
8. Structure of any field [103] --
9. The number of roots of a polynomial in a field [104] --
10. Polynomials in several elements [105] --
11. Symmetric polynomials [107] --
12. Rings of functions [110] --
CHAPTER IV: ELEMENTARY FACTORIZATION THEORY --
1. Factors, associates, irreducible elements [114] --
2. Gaussian semi-groups [115] --
3. Greatest common divisors [118] --
4. Principal ideal domains [121] --
5. Euclidean domains [122] --
6. Polynomial extensions of Gaussian domains [124] --
CHAPTER v: GROUPS WITH OPERATORS --
1. Definition and examples of groups with operators [128] --
2. M-subgroups, M-factor groups and M-homomorphisms [130] --
3. The fundamental theorem of homomorphism for M-groups [132] --
4. The correspondence between M-subgroups determined by a homomorphism [133] --
5. The isomorphism theorems for M-groups [135] --
6. Schreier’s theorem [137] --
7. Simple groups and the Jordan-Holder theorem [139] --
8. The chain conditions [142] --
9. Direct products [144] --
10. Direct products of subgroups [145] --
11. Projections [149] --
12. Decomposition into indecomposable groups [152] --
13. The Krull-Schmidt theorem [154] --
14. Infinite direct products [159] --
CHAPTER VI: MODULES AND IDEALS --
1. Definitions [162] --
2. Fundamental concepts [164] --
3. Generators. Unitary modules [166] --
4. The chain conditions [168] --
5. The Hilbert basis theorem [170] --
6. Noetherian rings. Prime and primary ideals [172] --
7. Representation of an ideal as intersection of primary ideals [175] --
8. Uniqueness theorems [177] --
9. Integral dependence [181] --
10. Integers of quadratic fields [184] --
CHAPTER VII: LATTICES --
1. Partially ordered sets [187] --
2. Lattices [189] --
3. Modular lattices [193] --
4. Schreier’s theorem. The chain conditions [197] --
5. Decomposition theory for lattices with ascending chain condition [201] --
6. Independence [202] --
7. Complemented modular lattices [205] --
8. Boolean algebras [207] --
Index [213] --
MR, MR0041102
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