Author | Jacobson, Nathan. author |
---|---|
Title | Lectures in Abstract Algebra I [electronic resource] : Basic Concepts / by Nathan Jacobson |
Imprint | New York, NY : Springer New York, 1951 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-7301-8 |
Descript | 217 p. online resource |
Introduction: Concepts from Set Theory the System of Natural Numbers -- 1. Operations on sets -- 2. Product sets, mappings -- 3. Equivalence relations -- 4. The natural numbers -- 5. The system of integers -- 6. The division process in I -- I: Semi-groups and Groups -- 1. Definition and examples of semi-groups -- 2. Non-associative binary compositions -- 3. Generalized associative law. Powers -- 4. Commutativity -- 5. Identities and inverses -- 6. Definition and examples of groups -- 7. Subgroups -- 8. Isomorphism -- 9. Transformation groups -- 10. Realization of a group as a transformation group -- 11. Cyclic groups. Order of an element -- 12. Elementary properties of permutations -- 13. Coset decompositions of a group -- 14. Invariant subgroups and factor groups -- 15. Homomorphism of groups -- 16. The fundamental theorem of homomorphism for groups -- 17. Endomorphisms, automorphisms, center of a group -- 18. Conjugate classes -- II: Rings, Integral Domains and Fields -- 1. Definition and examples -- 2. Types of rings -- 3. Quasi-regularity. The circle composition -- 4. Matrix rings -- 5. Quaternions -- 6. Subrings generated by a set of elements. Center -- 7. Ideals, difference rings -- 8. Ideals and difference rings for the ring of integers -- 9. Homomorphism of rings -- 10. Anti-isomorphism -- 11. Structure of the additive group of a ring. The charateristic of a ring -- 12. Algebra of subgroups of the additive group of a ring. One-sided ideals -- 13. The ring of endomorphisms of a commutative group -- 14. The multiplications of a ring -- III: Extensions of Rings and Fields -- 1. Imbedding of a ring in a ring with an identity -- 2. Field of fractions of a commutative integral domain -- 3. Uniqueness of the field of fractions -- 4. Polynomial rings -- 5. Structure of polynomial rings -- 6. Properties of the ring U[x] -- 7. Simple extensions of a field -- 8. Structure of any field -- 9. The number of roots of a polynomial in a field -- 10. Polynomials in several elements -- 11. Symmetric polynomials -- 12. Rings of functions -- IV: Elementary Factorization Theory -- 1. Factors, associates, irreducible elements -- 2. Gaussian semi-groups -- 3. Greatest common divisors -- 4. Principal ideal domains -- 5. Euclidean domains -- 6. Polynomial extensions of Gaussian domains -- V: Groups with Operators -- 1. Definition and examples of groups with operators -- 2. M-subgroups, M-factor groups and M-homomorphisms -- 3. The fundamental theorem of homomorphism for M-groups -- 4. The correspondence between M-subgroups determined by a homomorphism -- 5. The isomorphism theorems for M-groups -- 6. Schreierโs theorem -- 7. Simple groups and the Jordan-Hรถlder theorem -- 8. The chain conditions -- 9. Direct products -- 10. Direct products of subgroups -- 11. Projections -- 12. Decomposition into indecomposable groups -- 13. The Krull-Schmidt theorem -- 14. Infinite direct products -- VI: Modules and Ideals -- 1. Definitions -- 2. Fundamental concepts -- 3. Generators. Unitary modules -- 4. The chain conditions -- 5. The Hilbert basis theorem -- 6. Noetherian rings. Prime and primary ideals -- 7. Representation of an ideal as intersection of primary ideals -- 8. Uniqueness theorems -- 9. Integral dependence -- 10. Integers of quadratic fields -- VII: Lattices -- 1. Partially ordered sets -- 2. Lattices -- 3. Modular lattices -- 4. Schreierโs theorem. The chain conditions -- 5. Decomposition theory for lattices with ascending chain condition -- 6. Independence -- 7. Complemented modular lattices -- 8. Boolean algebras