Algebra: Volume I

 Content: 12 Linear Algebra.- 12.1 Modules over a Ring.- 12.2 Modules over Euclidean Rings. Elementary Divisors.- 12.3 The Fundamental Theorem of Abelian Groups.- 12.4 Representations and Representation Modules.- 12.5 Normal Forms of a Matrix in a Commutative Field.- 12.6 Elementary Divisors and Characteristic Functions.- 12.7 Quadratic and Hermitian Forms.- 12.8 Antisymmetric Bilinear Forms.- 13 Algebras.- 13.1 Direct Sums and Intersections.- 13.2 Examples of Algebras.- 13.3 Products and Crossed Products.- 13.4 Algebras as Groups with Operators. Modules and Representations.- 13.5 The Large and Small Radicals.- 13.6 The Star Product.- 13.7 Rings with Minimal Condition.- 13.8 Two-Sided Decompositions and Center Decomposition.- 13.9 Simple and Primitive Rings.- 13.10 The Endomorphism Ring of a Direct Sum.- 13.11 Structure Theorems for Semisimple and Simple Rings.- 13.12 The Behavior of Algebras under Extension of the Base Field.- 14 Representation Theory of Groups and Algebras.- 14.1 Statement of the Problem.- 14.2 Representation of Algebras.- 14.3 Representations of the Center.- 14.4 Traces and Characters.- 14.5 Representations of Finite Groups.- 14.6 Group Characters.- 14.7 The Representations of the Symmetric Groups.- 14.8 Semigroups of Linear Transformations.- 14.9 Double Modules and Products of Algebras.- 14.10 The Splitting Fields of a Simple Algebra.- 14.11 The Brauer Group. Factor Systems.- 15 General Ideal Theory of Commutative Rings.- 15.1 Noetherian Rings.- 15.2 Products and Quotients of Ideals.- 15.3 Prime Ideals and Primary Ideals.- 15.4 The General Decomposition Theorem.- 15.5 The First Uniqueness Theorem.- 15.6 Isolated Components and Symbolic Powers.- 15.7 Theory of Relatively Prime Ideals.- 15.8 Single-Primed Ideals.- 15.9 Quotient Rings.- 15.10 The Intersection of all Powers of an Ideal.- 15.11 The Length of a Primary Ideal. Chains of Primary Ideals in Noetherian Rings.- 16 Theory of Polynomial Ideals.- 16.1 Algebraic Manifolds.- 16.2 The Universal Field.- 16.3 The Zeros of a Prime Ideal.- 16.4 The Dimension.- 16.5 Hilbert\'s Nullstellensatz. Resultant Systems for Homogeneous Equations.- 16.6 Primary Ideals.- 16.7 Noether\'s Theorem.- 16.8 Reduction of Multidimensional Ideals to Zero-Dimensional Ideals.- 17 Integral Algebraic Elements.- 17.1 Finite R-Modules.- 17.2 Integral Elements over a Ring.- 17.3 The Integral Elements of a Field.- 17.4 Axiomatic Foundation of Classical Ideal Theory.- 17.5 Converse and Extension of Results.- 17.6 Fractional Ideals.- 17.7 Ideal Theory of Arbitrary Integrally Closed Integral Domains.- 18 Fields with Valuations.- 18.1 Valuations.- 18.2 Complete Extensions.- 18.3 Valuations of the Field of Rational Numbers.- 18.4 Valuation of Algebraic Extension Fields: Complete Case.- 18.5 Valuation of Algebraic Extension Fields: General Case.- 18.6 Valuations of Algebraic Number Fields.- 18.7 Valuations of a Field ?(x) of Rational Functions.- 18.8 The Approximation Theorem.- 19 Algebraic Functions of One Variable.- 19.1 Series Expansions in the Uniformizing Variable.- 19.2 Divisors and Multiples.- 19.3 The Genus g.- 19.4 Vectors and Covectors.- 19.5 Differentials. The Theorem on the Speciality Index.- 19.6 The Riemann-Roch Theorem.- 19.7 Separable Generation of Function Fields.- 19.8 Differentials and Integrals in the Classical Case.- 19.9 Proof of the Residue Theorem.- 20 Topological Algebra.- 20.1 The Concept of a Topological Space.- 20.2 Neighborhood Bases.- 20.3 Continuity. Limits.- 20.4 Separation and Countability Axioms.- 20.5 Topological Groups.- 20.6 Neighborhoods of the Identity.- 20.7 Subgroups and Factor Groups.- 20.8 T-Rings and Skew T-Fields.- 20.9 Group Completion by Means of Fundamental Sequences.- 20.10 Filters.- 20.11 Group Completion by Means of Cauchy Filters.- 20.12 Topological Vector Spaces.- 20.13 Ring Completion.- 20.14 Completion of Skew Fields.