Lectures in Abstract Algebra I

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. Schreier's theorem.- 7. Simple groups and the Jordan-Hölder 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. Schreier's theorem. The chain conditions.- 5. Decomposition theory for lattices with ascending chain condition.- 6. Independence.- 7. Complemented modular lattices.- 8. Boolean algebras.