INTRODUCTION: CONCEPTS FROM SET THEORY THE SYSTEM OF NATURAL NUMBERS
SECTION
1.Operationsonsets
2.Product sets, mappings
3.Equivalencerelations
4.Thenaturalnumbers
5.Thesystemofintegers
6.The division process in I
CHAPTER I: SEMI-GROUPS AND GROUPS
1.Definition and examples ofsemi-groups
2.Non-associative binary compositions
3.Generalized associativelaw.Powers
4.Commutativity
5.Identities andinverses
6.Definition and examples of groups
7.Subgroups
8.Isomorphism
9.Transformation groups
10.Realization of a group as a transformation group
II.Cyclic groups.Order of an element
12.Elementary properties ofpermutations
13.Coset decompositions ofa group
14.Invariant subgroups and factor groups
15.Homomorphismofgroups
16.The fundamental theorem of homomorphism for groups
17.Endomorphisms, automorphisms, center of a group
18.Conjugatc classes
CHAPTER II: RINGS, INTEGRAL DOMAINS AND FIELDS
SECTION
1.Definition andexamples
2.Typesofrings
3.Quasi-regularity.The circle composition
4.Matrixrings
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 ofrings
10.Anti-isomorphism
11.Structure of the additive group of a ring.The charateristic ofaring
12.Algebra of subgroups of the additive group of a ring.Onr sidedideals
13.The ring of endomorphisms of a commutative group
14.The multiplications of a ring
CHAPTER 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.Polynomialrings
5.Structure of polynomial rings
6.Properties of the ring 2l[x]
7.Simple extensions ofa field
8.Structureofany field
9.The number of roots of a'polynomial in a field
10.Polynomials in several elements
11.Symmetric polynomials
12.Ringsoffunctions
CHAPTER IV: ELEMENTARY FACTORIZATlON THEORY
1.Factors, associates, irreducible elements
2.Gaussian semi-groups
3.Greatest common divisors
4.Principalidealdomains
SECTION
5.Euclidean domains
6.Polynomial extensions of Gaussian domains
CHAPTER 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-HSlder 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
CHAPTER VII 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 ideal
8.Uniqueness theorems
9.Integral dependence
10.Integers of quadratic fields
CHAPTER VII: LATTICES
1.Partially ordered sets
2.Lattices
3.Modular lattices
4.Schreier's theorem.The chain condition
SECTION
5.Decomposition theory for lattices with ascending chain condition
6.Independence
7.Complemented modular lattices
8.Boolean algebras
Index
