#algebra
20. Group Theory and Generalizations
This volume studies groups as algebraic structures encoding symmetry. It develops structural, combinatorial, and computational aspects, along with generalizations such as group actions and extensions. Part I. Foundations of Group Theory Chapter 1. Groups 1.1 Definitions and examples 1.2 Subgroups 1.3 Cyclic groups 1.4 Permutation groups 1.5 Basic properties Chapter 2. Homomorphisms 2.1 Group homomorphisms 2.2 Kernels and images 2.3 Isomorphism theorems 2.4 Automorphisms 2.5 Examples Chapter 3. Cosets and...
19. K-Theory
This volume studies algebraic and topological K-theory, focusing on invariants derived from vector bundles, modules, and operator algebras. It provides tools for classification problems across algebra, topology, and geometry. Part I. Foundations of K-Theory Chapter 1. Motivation and Basic Definitions 1.1 Classification problems 1.2 Grothendieck groups 1.3 Additive invariants 1.4 Exact sequences 1.5 Examples Chapter 2. Grothendieck Construction 2.1 Monoids to groups 2.2 Universal properties 2.3 Functoriality 2.4 Examples in...
17. Non-Associative Rings and Algebras
This volume studies algebraic systems where associativity does not hold in general. It includes Lie algebras, Jordan algebras, alternative algebras, and related structures. These systems arise naturally in geometry, physics, and symmetry theory. Part I. Foundations Chapter 1. Non-Associative Structures 1.1 Definitions and examples 1.2 Binary operations without associativity 1.3 Identities and laws 1.4 Homomorphisms 1.5 Subalgebras and quotients Chapter 2. Basic Identities 2.1 Associator and commutator 2.2 Flexible and...
14. Algebraic Geometry
This volume studies geometric objects defined by polynomial equations. It connects commutative algebra, geometry, and number theory through the language of varieties and schemes. Part I. Affine Algebraic Geometry Chapter 1. Affine Varieties 1.1 Polynomial rings and ideals 1.2 Zero sets of polynomials 1.3 Affine space 1.4 Zariski topology 1.5 Examples Chapter 2. Coordinate Rings 2.1 Functions on varieties 2.2 Radical ideals 2.3 Nullstellensatz 2.4 Correspondence between ideals and varieties...
16. Associative Rings and Algebras
This volume studies rings and algebras with associative multiplication, without requiring commutativity. It develops structure theory, module theory, and connections to representation theory and geometry. Part I. Basic Ring Theory Chapter 1. Rings and Homomorphisms 1.1 Definitions and examples 1.2 Subrings and ideals 1.3 Ring homomorphisms 1.4 Quotient rings 1.5 Basic constructions Chapter 2. Ideals and Structure 2.1 Left, right, and two-sided ideals 2.2 Prime and maximal ideals 2.3 Jacobson...
12. Field Theory and Polynomials
This volume studies fields, polynomials, and algebraic extensions. It provides the structural basis for algebraic number theory, algebraic geometry, and Galois theory. Part I. Fields and Basic Structures Chapter 1. Fields 1.1 Definition and examples 1.2 Subfields 1.3 Field homomorphisms 1.4 Characteristic of a field 1.5 Prime fields Chapter 2. Polynomial Rings 2.1 Polynomial definitions 2.2 Arithmetic of polynomials 2.3 Degree and leading coefficient 2.4 Division algorithm 2.5 Euclidean structure...
13. Commutative Algebra
This volume studies commutative rings, ideals, modules, and their structural properties. It forms the algebraic foundation for algebraic geometry and number theory. Part I. Rings and Ideals Chapter 1. Commutative Rings 1.1 Definitions and examples 1.2 Ring homomorphisms 1.3 Subrings and quotient rings 1.4 Units and zero divisors 1.5 Basic constructions Chapter 2. Ideals 2.1 Definition and examples 2.2 Prime and maximal ideals 2.3 Operations on ideals 2.4 Ideal quotient...
15. Linear and Multilinear Algebra; Matrix Theory
This volume develops vector spaces, linear maps, matrices, and multilinear structures. It serves as a core toolkit for nearly all areas of mathematics, physics, and computation. Part I. Vector Spaces Chapter 1. Vector Spaces 1.1 Definitions and examples 1.2 Subspaces 1.3 Linear combinations 1.4 Span and generation 1.5 Linear independence Chapter 2. Bases and Dimension 2.1 Bases 2.2 Existence of bases 2.3 Dimension 2.4 Coordinate representations 2.5 Change of basis...
08. General Algebraic Systems
This volume studies algebraic systems in their most abstract form. It focuses on operations, identities, and structures without restricting to specific classes like groups or rings, providing a unifying framework for all algebraic theories. Part I. Universal Algebra Foundations Chapter 1. Algebraic Structures 1.1 Sets with operations 1.2 Signatures and arities 1.3 Terms and term algebras 1.4 Homomorphisms 1.5 Subalgebras and products Chapter 2. Identities and Equations 2.1 Equational logic...
11. Number Theory
This volume studies integers and arithmetic structures. It covers divisibility, primes, modular arithmetic, Diophantine equations, and modern analytic and algebraic methods. Part I. Integers and Divisibility Chapter 1. Basic Properties of Integers 1.1 Divisibility and factors 1.2 Prime numbers 1.3 Greatest common divisor 1.4 Euclidean algorithm 1.5 Fundamental theorem of arithmetic Chapter 2. Arithmetic Functions 2.1 Definition and examples 2.2 Multiplicative functions 2.3 Möbius function 2.4 Euler totient function 2.5...
06. Order and Lattices
This volume studies partially ordered sets, lattices, and algebraic systems equipped with order relations. It connects combinatorics, algebra, topology, and logic through the notion of order. Part I. Foundations of Order Theory Chapter 1. Partially Ordered Sets 1.1 Definitions: posets, relations 1.2 Reflexivity, antisymmetry, transitivity 1.3 Hasse diagrams 1.4 Chains and antichains 1.5 Examples and constructions Chapter 2. Basic Properties 2.1 Minimal and maximal elements 2.2 Bounds and intervals 2.3...