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 ↓


Free Modules


In algebra, modules are mathematical structures that generalize notions of vector spaces. A module over a ring allows us to extend linear algebra to a wider framework over fields. Free modules are an essential concept in module theory. They have a basis, much like free vector spaces, and allow us to understand many structural properties of modules over rings.

Understanding the module

Before diving into free modules, let's first understand what modules are. Modules are a generalization of the idea of a vector space. However, in a vector space, the scalars come from fields, while in a module, the scalars come from rings. This difference allows modules to be used in a wide range of applications.

Definition: A module M over a ring R is an abelian group with an operation · : R x M → M such that for all m, n ∈ M, and a, b ∈ R: 1. a · (m + n) = (a · m) + (a · n) 2. (a + b) · m = (a · m) + (b · m) 3. (a · b) · m = a · (b · m) 4. 1_R · m = m

Free modules

A free module is similar to a vector space that has a free and spanning set. We call this set a basis, and each element in a free module can be uniquely expressed as a finite linear combination of elements from the basis. Let's see what this means in detail.

Definition: A module F is said to be free if there exists a set { e i } for i in some index set I such that every element f in F can be uniquely written as: f = Σ a i e i where a i are elements from the ring R, and almost all a i are zero (ie, for finitely many i in I, a i is non-zero).

Examples of free modules

Let us consider some examples to strengthen this concept.

Example 1: The space Rn as a module over R

Let's take the familiar R n, which is a free module over the ring R (the real numbers), which is the same as being a vector space. The standard basis is {e 1, e 2, ..., en}, where:

e 1 = (1, 0, 0, ..., 0), e 2 = (0, 1, 0, ..., 0), ... e n = (0, 0, 0, ..., 1)

Any element v = (v 1, v 2, ..., v n) in R n can be written as:

v = v 1 e 1 + v 2 e 2 + ... + v n e n

Example 2: Free modules over the integers

Consider the integers Z as a module over themselves. Z is a free Z-module over the basis {1}. Any integer n can be expressed as:

n = n · 1

Here, the endomorphism ring of the free module is isomorphic to the ring itself.

Visual representation of free modules

Basis vector E1 Basis vector e 2

The above diagram represents a free module over a 2-component system, with the red dots indicating the basis vectors, thereby spanning the module space.

Properties of free modules

Free modules have several interesting properties:

  • Existence of a basis: every free module has a basis, and this basis is important for uniquely expressing any module element.
  • Uniqueness of representation: every element of a free module can be uniquely represented as a sum of elements from its basis.
  • Homeomorphism: Any homeomorphism from a free module to any module is determined only by what happens in the basis.

Example of homeomorphism

Consider a free module F with basis {e 1, e 2}. Any homomorphism φ from F to another module M can be uniquely determined by specifying φ(e 1) and φ(e 2).

Free vs. non-free modules

Not all modules are free. To understand this, consider a module over a ring under certain conditions. A module over a ring may have no basis, which makes it non-independent.

Example of a non-free module

Consider the Z-module (module over the integers) Z/2Z. This module is not free because it cannot have a basis that spans the entire set with a unique representation.

While free modules look like simple, vector-like structures, non-free modules require dealing with torsion elements and other complications.

Free module creation

Free modules can be constructed in various ways depending on the starting point, often starting with a set and a ring.

Method: free modules generated by a set

Let S be a set and R a ring. The free R-module generated by S is composed of formal expressions of the form:

f = Σ r i s i where r i ∈ R, s i ∈ S, and almost all r i are zero.

The collection of these expressions forms a free module over S as a basis.

Applications of free modules

Free modules are important in several mathematical areas:

  • Algebraic geometry: free modules possibly help in understanding sheaves of modules over complex rings.
  • Homology theory: Chain complexes in homology are often constructed using free modules.
  • Computational group theory: analyzing permutation groups often involves studying free modules over the group elements.

Conclusion

Free modules provide a straightforward but profound way to extend linear algebra beyond fields, including more general algebraic structures and more explicit tools for mathematical exploration. When defining the basis and representation aspects of modules over rings, free modules form an essential concept in both theoretical and practical contexts, shedding light on various properties associated with module theory.


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Free Modules

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