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 ↓


Modules over Rings


In the field of algebra, module theory extends the familiar concept of a vector space by replacing the field of scalars with a more general ring. This generalization unveils a world of structures and properties that capture many algebraic phenomena.

Introduction

Modules over rings are an important concept in abstract algebra. While vector spaces are built over fields, modules are built over rings. This simple difference opens up vast possibilities and applications in mathematics. A module provides a framework for understanding structures that are central not only in algebra but also in topology, geometry, and other mathematical domains.

Definitions and basics

Before diving into modules, we need to understand what rings are. A ring is a set equipped with two operations, typically addition and multiplication, where the set is an abelian group under addition and is associative under multiplication. Moreover, multiplication is distributive over addition.

A module over a ring is a generalization of the concept of a vector space. Specifically, a module over a ring ( R ) is an additive abelian group ( M ) equipped with an operation that associates to each element ( r in R ) and ( m in M ) an element ( rm in M ), satisfying some properties similar to scalar multiplication in vector spaces.

Formal definition

If ( R ) is a ring, then an R-module is an abelian group ( (M, +, 0) ) together with an operation ( R times M to M ) (called scalar multiplication) denoted ( (r, m) mapsto rm ) such that for all ( r, s in R ) and ( m, n in M ):

1. ( r cdot (m + n) = r cdot m + r cdot n ) (distributive over module addition) 2. ( (r + s) cdot m = r cdot m + s cdot m ) (distributive over ring addition) 3. ( (rs) cdot m = r cdot (s cdot m) ) (associativity) 4. ( 1_R cdot m = m ) if the ring has a multiplicative identity 1.

Simple examples of modules

Example 1: Vector space

The most familiar example of a module is a vector space over a field ( F ). In this case, if the ring ( R ) is a field ( F ), then the modules are properly vector spaces over ( F ).

Example 2: ( mathbb{Z} )-module

Any abelian group can be viewed as a module over the ring of integers ( mathbb{Z} ). The action is defined by ( n cdot a = a + a + cdots + a ) (n times) for ( n in mathbb{Z} ) and ( a in M ).

Example 3: Matrices

Consider a ring of ( n times n ) matrices with coefficients in a ring ( R ). The set of all ( n times 1 ) column vectors of size ( n ) with coefficients in an ( R )-module forms an ( R )-module. Matrix multiplication acts like scalar multiplication in this module.

Visualizing module

To conceptualize modules visually, think of a set of objects (such as vectors) on which you can perform operations such as addition between these objects and multiplication by an outer set of elements (from the ring). Consider the following representations of operations in modules.

M = , ,

In this simple illustration, the elements of the module ( M ) are represented as colored dots, and their sum results in another element in the module. Scalar multiplication can be depicted similarly by changing the positions of the elements (or by scaling).

Submodules and quotient modules

Similar to subspaces in a vector space, modules can have submodules. A subgroup ( N subseteq M ) is a submodule if it is closed under addition and scalar multiplication. Quotient modules provide a way to 'compact' or factorize through certain submodules, giving rise to new module structures.

Example of submodules

Take ( M = mathbb{Z} ) as a ( mathbb{Z} )-module. Any ( nmathbb{Z} ), where ( n ) is an integer, forms a submodule since it is closed under addition and multiplication by integers.

The module ( mathbb{Z}_n = mathbb{Z}/nmathbb{Z} ) is the quotient module constructed from ( mathbb{Z} ) by the submodule ( nmathbb{Z} ). It corresponds to ordinary arithmetic modulo ( n ).

Homeomorphism of modules

A homomorphism between two ( R )-modules ( M ) and ( N ) is a function ( f: M to N ) that respects the operation of modules:

1. Additivity: ( f(m + n) = f(m) + f(n) ) for all ( m, n in M ) 2. Scalar multiplication: ( f(r cdot m) = r cdot f(m) ) for all ( r in R ) and ( m in M ).

These homeomorphisms play a role similar to linear transformations in vector space theory. They act as 'algebraic maps' between modules while preserving the module structure.

Example of homeomorphism

Consider ( mathbb{Z} )-modules ( mathbb{Z} ) and ( mathbb{Z}_n ). The map ( f: mathbb{Z} to mathbb{Z}_n ) defined by ( f(a) = a mod n ) is a module homomorphism.

Direct sum of modules

Given a module ( M ) with two submodules ( N_1 ) and ( N_2 ) such that every element ( m in M ) can be uniquely written as ( m = n_1 + n_2 ) for ( n_1 in N_1 ) and ( n_2 in N_2 ), the module ( M ) is the direct sum of ( N_1 ) and ( N_2 ) which is denoted as ( M = N_1 oplus N_2 ).

For example, consider ( mathbb{Z}_6 ). As a ( mathbb{Z} )-module, it can be decomposed into a direct sum of its submodules for various divisors.

Challenges in module theory

One of the complications of modules over rings, unlike vector spaces over fields, is that many of the properties familiar with vector spaces do not hold. For example, not every module is free (has a basis), and many rings have no trivial or unique factorization into 'simplest' parts.

Modules over non-commutative rings introduce further complications, since left modules and right modules can behave quite differently depending on the structure of the rings.

Applications of module theory

Modules appear in a variety of mathematical contexts, from the study of abelian groups (viewed as ( mathbb{Z} )-modules) to the solution of systems of linear equations over rings. They also play a foundational role in the study of algebraic geometry and topology, via rings of functions on geometric objects.

For example, in algebraic geometry, coherent sheafs of various types correspond to modules over the ring of sections of the sheaf. This connects algebraic properties of modules to geometric properties of the underlying space.

Conclusion

Modules over rings embody a rich algebraic structure that extends far beyond the familiar boundaries of vector spaces. Although they can introduce additional complexities and challenges, the generality of module theory allows for a deeper exploration and understanding of algebraic systems. Modules prove indispensable in connecting different branches of mathematics and expanding the range of algebraic application.


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Modules over Rings

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