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 → Ring Theory ↓


Unique Factorization Domains


Unique factorization domains (UFDs) are an important concept in ring theory, a branch of abstract algebra. In mathematics, particularly ring theory, a unique factorization domain is a commutative ring in which every non-zero and non-unit element can be written as a product of indecomposable elements (or "atoms"), so that this representation is unique up to order and units. This concept generalizes the well-known fundamental theorem of arithmetic, which states that every integer greater than one can be uniquely represented as a product of prime numbers.

Introduction to rings

Before delving into unique factorization domains, it is important to understand the fundamental concept of rings in mathematics:

  • Definition of a ring: A ring is a set equipped with two binary operations: addition and multiplication, which we usually denote by + and *. A ring allows us to perform the following operations:
    • Add any two elements together to get another element in the ring.
    • Multiply any two elements together to get another element in the ring.
    • Include a zero element, which is the additive identity, meaning that for any element a in the ring, a + 0 = a.
    • It contains an element which when multiplied by itself gives the same element, although in some rings such an element is not necessarily present.

Examples of rings

Here are some examples to illustrate the rings:

  • The set of integers Z, together with the usual operations of addition and multiplication, forms a ring.
  • Consider the set of even integers under addition and multiplication. This forms a ring because adding or multiplying two even numbers gives another even number.

Integral domain

In rings, certain types of subgroups are particularly important. Integral domains are a special kind of ring:

  • Definition: An integral domain is a commutative ring that has unity (1 ≠ 0), and no zero divisors.
  • Properties of Integral Domains: An integral domain ensures that if ab = 0 for two elements a and b in the domain, then either a = 0 or b = 0.

An example of an integral domain is the set Z of all integers.

What are the typical factorization domains?

A unique factorization domain (UFD) is based on the properties of integral domains:

  • Definition: A UFD is a type of integral domain in which every element a ≠ 0 can be partitioned into indecomposable elements, and this partition is unique.
  • Irreversible elements: These are elements that cannot be factored into more than one factor other than one times themselves.

Fundamentals of arithmetic

Before we proceed further, let's revisit a familiar concept called the "fundamental theorem of arithmetic," which states that every integer greater than 1 is either a prime number, or can be formulated as a unique product of prime numbers, regardless of the order of multiplication.

        30 = 2 * 3 * 5
        60 = 2^2 * 3 * 5

This theorem is directly related to the concept of unique factorization domains by showing that there is a simplification of unique factorization in the domain of integers Z

Specification of factorization

In a UFD, uniqueness means that you can rearrange factors or factor out units (elements whose multiplication is in the inverse set), but no completely distinct set of invariant factors can be found. Let's demonstrate this with integers, which form a UFD:

        12 = 2 * 2 * 3 (These are the only invariant factors up to order and units.)

For example, the number 12 can only be factored into 2 * 2 * 3 or rearranged as 3 * 2 * 2, which shows the unique factorization property of integers.

Polynomials and UFDs

Polynomials form another important class of UFDs:

  • The ring of polynomials with a single indeterminant over the integers, denoted Z[x], is a UFD.
  • Consider the polynomial: f(x) = 2x^2 + 4x + 2.
  • This polynomial can be factored as f(x) = 2(x + 1)^2, which indicates factorization into irreducible elements (in this case, the factors cannot be simplified any further).

Visual representation

Let us understand the concept of factorization. Below, we can illustrate the factorization of the number 30, which can be uniquely factored into prime numbers:

30 2 3 5

In this picture, rectangular boxes represent numbers. The top box, labeled 30, is divided into three factors shown in green: 2, 3, and 5.

Distinctive properties of UFD

Understanding the distinctive properties of UFDs helps to enhance understanding:

  • Associative: Two elements are considered associative if each divides the other. In a UFD, the invariant factors are unique up to these associatives.
  • Principal Ideal Domain (PID): A UFD is always a principal ideal domain. In a PID, each ideal is generated by a single element.

That ring can't be a UFD

Not all rings are UFDs. Consider the ring Z[√-5], which is the set of all numbers of the form a + b√-5, where a and b are integers. This sub-ring is not a UFD because some numbers in this ring may have multiple distinct factors:

        6 = 2 * 3 = (1 + √-5)(1 - √-5)

The number 6 can be uniquely divided among the integers in this ring, but not in this extension, so it fails to qualify as a UFD.

Demonstrations with polynomials

Another example is the ring of polynomials with rational coefficients Q[x], which is a UFD.

        f(x) = (x^2 - 1) = (x - 1)(x + 1)

Here, the factorization for the polynomial f(x) is unique.

Conclusion: The importance of UFD

Unique factorization domains play a key role in simplifying algebraic structures and solving equations. They underlie many essential concepts in mathematics, ranging from number theory to algebraic geometry. Recognizing UFDs allows mathematicians to inform rigorous proofs, develop new theorems, and solve complex problems with the assurance of the uniqueness of the factorization.

In summary, UFDs extend the fundamental theorem of arithmetic to a wider range of mathematical objects beyond integers, extend important properties, ensure unique factorization, and provide better understanding and manipulation of algebraic entities.


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