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


Ring theory is a part of abstract algebra. It is the study of rings, which are algebraic structures equipped with two binary operations: addition and multiplication. Ring theory is important to understand because it has profound implications in various fields of mathematics and science. This discourse is designed to go deep into the theory, providing comprehensive information about what rings are, their properties, types, and applications.

Basic definition and properties

A ring is a set R equipped with two operations, commonly called addition and multiplication, that satisfy certain axioms:

Axioms

  • Additive closure: for all a, b in R, the sum a + b is also in R
  • Additive associativity: For all a, b, c in R, (a + b) + c = a + (b + c)
  • Additive identity: There exists an element 0 in R such that for all a in R, a + 0 = a.
  • Additive inverse: for every a in R, there exists an element -a in R such that a + (-a) = 0.
  • Multiplicative closure: for all a, b in R, the product a * b is in R
  • Multiplicative associativity: For all a, b, c in R, (a * b) * c = a * (b * c)
  • Distributive rules:
    • a * (b + c) = a * b + a * c
    • (a + b) * c = a * c + b * c

Unlike fields, rings are not required to have multiplicative inverses or commutative multiplication.

Visualization of the rings

To better understand rings, let's look at the basic operations and properties. Below is a simple illustration of a ring.

0

The figure above shows the additive identity of a ring. Adding any number to zero in the ring gives that number itself.

Example 1: Integer

The group of integers Z is a classic example of a ring. Let's investigate these ring properties using integers:

  • Additive closure: Any integers a and b result in an integer a + b.
  • Multiplicative closure: Any integers a and b give an integer a * b.

Types of rings

There are different types of rings, which are mainly classified based on their specific properties:

Interchangeable rings

A ring is commutative if the multiplication operation is commutative, that is, for all a, b in the ring, a * b = b * a. A classic example is the set of integers.

Unit rings and division rings

A unit ring, also called a ring with unity, has an element 1 such that for all a in the ring, a * 1 = a. A division ring allows division by non-zero elements, but multiplication is not necessarily commutative, as in the case of quaternions.

Visualization through diagrams

Looking at the different types of rings can help illustrate their structure:

Interchangeable Unit

More ring properties and examples

Norms and sub-rings

There are some subgroups of rings that behave like rings themselves:

  • Sub-rings: A subset of a ring that forms a ring with the same operations.
  • Ideal: A subset I of a ring R where for every a in I and every r in R, both ra and ar are in I

Polynomial rings

Polynomial rings are built from a ring by taking coefficients, usually real numbers or integers, and using polynomial expressions. They obey the ring properties with polynomial addition and multiplication.

Applications of ring theory

Ring theory is fundamental to many advanced mathematical fields such as algebraic geometry, number theory, and cryptography. For example, polynomial rings form the backbone of coding theory, helping to implement error detection and correction.

Example application

Consider polynomials with integer coefficients, which form a ring, Z[x] These are important in computer science for tasks such as building efficient error-checking algorithms.

    P(x) = 2x^2 + 3x + 1 ∈ Z[x] Q(x) = 4x - 5 ∈ Z[x] Sum: (P + Q)(x) = 2x^2 + 7x - 4 Product: (P * Q)(x) = 8x^3 + 2x^2 - 17x - 5

Conclusion

Ring theory is not just about understanding the operations within a set. It opens up a vast world where algebraic structures combine with logic, providing avenues for finding solutions to seemingly impossible problems. From basic integer operations to complex polynomial rings used in technology, ring theory underlies the essence of abstract algebra. Embracing its intricacies enhances our overall understanding and leads to success in mathematical subjects.


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Ring Theory

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