PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Algebra ↓


Module Theory


In mathematics, module theory is a fascinating study that extends concepts from linear algebra and introduces additional complexity and richness. It is a branch of abstract algebra that deals with modules, which generalize vector spaces by allowing the set of scalars to be rings instead of fields. Let's understand module theory in depth with some simple language and visual examples.

Inspiration and basics

In linear algebra, we often work with vector spaces. A vector space is a collection of vectors that can be added together and multiplied by numbers (scalars). These scalars come from a field, such as the rational numbers (ℚ), the real numbers (ℝ), or the complex numbers (ℂ).

However, sometimes we want to work with scalars that are not in a field, such as the integers (ℤ) or polynomials. This broader context leads us to modules. Modules are like vector spaces, except that they allow scalars to come from rings rather than fields.

Definition of module

Modules are a generalization of the concept of vectors. Formally, assume R to be a ring. A module M over R is a set equipped with two operations:

  • Addition: +, which is commutative, associative, and has an identity element.
  • Scalar multiplication: for every r ∈ R and m ∈ M, there exists an element rm ∈ M such that the distributive laws apply with respect to both module addition and ring addition/multiplication.

Visual example

Consider a ring R and a module M over R This diagram shows that the elements of the module are acted upon by the elements of the ring:

R: ooooo (elements of the ring) | *-----> (operation) | M: • • • • • (elements of the module)

Simple example

Let us understand what modules are and how they work with some examples.

Example 1: Vector space as a module

The simplest example of a module is a vector space. Suppose we have a field F Then, a vector space V over F is a module where the ring R = F Thus, every vector space can be viewed as a module over its base field.

Example 2: ℤ-module

Consider ℤ (the group of all integers) as a ring. A module over ℤ is called an ℤ-module. Any abelian group can be thought of as an ℤ-module. For example, ℤⁿ, the group of all n tuples of integers, forms an ℤ-module under component-wise addition and scalar multiplication.

Example 3: Modules over polynomial rings

If we consider a ring R = ℝ[x] (the ring of all polynomials with real coefficients), then modules over this ring are called ℝ[x] -modules. An example is the set of all polynomials with real coefficients of a given degree n.

Homeomorphism of modules

A homomorphism between two R modules M and N is a function f: M → N that respects the module operation. In particular, for all u, v ∈ M and r ∈ R, the following must hold:

  • f(u + v) = f(u) + f(v) (additivity)
  • f(ru) = rf(u) (compatibility with scalar multiplication)

The set of all R module homomorphisms from M to N is denoted by Hom_R(M, N).

Submodules

Just as vector spaces have subspaces, modules have submodules. A submodule N of a module M is a subgroup that is closed under addition and scalar multiplication. More formally, if N ⊆ M, N is a submodule if:

  • 0 ∈ N (contains zero element)
  • u, v ∈ N implies u + v ∈ N (closure under addition)
  • r ∈ R and u ∈ N implies ru ∈ N (closure under scalar multiplication)

Quotient module

Given a module M and a submodule N, we can construct a quotient module M/N, which is similar to factor groups in group theory or quotient spaces in linear algebra. The elements of the quotient module are the cosets of N in M Addition and scalar multiplication are defined naturally as follows:

  • (u + N) + (v + N) = (u + v) + N
  • r(u + N) = (ru) + N

Finitely generated modules

A module M is called finitely generated if there exists a finite set of elements {m₁, m₂, ..., mₙ} in M such that every element m ∈ M can be expressed as a finite R -linear combination of these generators. Formally,

m = r₁m₁ + r₂m₂ + ... + rₙmₙ

Free modules

A free module is a module that has a basis, which is similar to the concept of a basis for a vector space. A module M is free if there exists a set {eᵢ} of elements such that each element m ∈ M can be written uniquely as a finite sum:

m = r₁e₁ + r₂e₂ + ... + rₖeₖ

where rᵢ ∈ R and the coefficient rᵢ is unique.

Visual example of free module

Imagine the module M as a collection of arrows (vectors) and each element e₁, e₂,... in the basis corresponds to a direction:

e₁: →→→→→ e₂: ↗↗↗↗↗ m = r₁e₁ + r₂e₂

Exact sequence

Exact sequences are tools used to understand module structures more deeply. An exact sequence is a sequence of module homomorphisms:

... → A → B → C → ...

This is exact on a module B if the image of the map before B is equal to the kernel of the map after B Exact sequences help us understand how modules are built up from each other.

Applications of module theory

  • Algebraic topology: modules are used in defining homology and cohomology groups of topological spaces.
  • Algebraic geometry: Module theory is used to study sheaves of modules and coherent sheaves.
  • Representation theory: modules are used to represent algebraic structures, such as groups and algebras.

Conclusion

Module theory is a rich and detailed area of algebra that has connections to many other areas of mathematics. It generalizes the concept of vector space and integrates ideas from ring theory, making it an essential tool for understanding complex algebraic structures. By understanding the basics outlined here, one can delve deeper into advanced topics and explore its vast applications in modern mathematics.


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Module Theory

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