Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Abstract Algebra ↓


Rings and Fields


In the study of abstract algebra, two important structures are often encountered, rings and fields. These structures arise from the desire to generalize the arithmetic we are familiar with, such as addition, subtraction, multiplication, and division, and apply it to a variety of mathematical systems. Understanding rings and fields helps advance mathematical theory and find practical applications in various branches of science and engineering.

Understanding the rings

A ring is a mathematical set equipped with two operations that generalize the operations of addition and multiplication. A ring is defined by the following properties:

  1. Closure under addition: if a and b are in a ring, then a + b is also in the ring.
  2. Associative sum: for all elements a, b and c in the ring, (a + b) + c = a + (b + c)
  3. Commutative addition: for every two elements a and b in the ring, a + b = b + a.
  4. Additive identity: there exists an element 0 in the ring such that a + 0 = a for every element a in the ring.
  5. Additive inverse: for every element a in the ring, there exists an element -a such that a + (-a) = 0.
  6. Closure under multiplication: if a and b are in a ring, then a * b is also in a ring.
  7. Associative multiplication: for all elements a, b, and c in the ring, (a * b) * c = a * (b * c)
  8. Distributive law: multiplication distributes over addition: a * (b + c) = (a * b) + (a * c) and (a + b) * c = (a * c) + (b * c) for all elements a, b, and c in the ring.

It is important to note that multiplication in a ring may not be commutative, and the ring need not have a multiplicative identity.

Example of a ring

Let us consider the set of integers (mathbb{Z}) with the usual operations of addition and multiplication. To check whether this forms a ring, we will verify the properties of a ring:

  • Closure under addition: The sum of any two integers is an integer.
  • Associative and commutative addition: Both properties are valid because integer addition behaves just like regular addition.
  • Additive identity and inverse: The number 0 serves as an identity element, and for any integer a, -a is also an integer.
  • Commutativity, Associativity and Distributive Laws in Multiplication: Similarly, the multiplication of integers satisfies these properties.

Here is a visual demonstration of forming a ring by integers:

Add Multiplication Closure

Area: Advanced structure of rings

Fields are extensions of rings that have the additional property that division is possible (except division by zero). Formally, a field is a ring with additional properties:

  1. Multiplicative commutativity: for every two elements a and b in the field, a * b = b * a.
  2. Multiplicative identity: there exists a non-zero element 1 such that a * 1 = a for every element a in the field.
  3. Multiplicative inverse: for every non-zero element a in the field, there exists an element a-1 such that a * a-1 = 1.

In a field, every non-zero element must have a multiplicative inverse, allowing division by non-zero elements.

Example of a field

The set of real numbers (mathbb{R}) with the standard operations of addition and multiplication is a field. Let's verify why:

  • Additive and Multiplicative Operations: Addition of real numbers satisfies all the properties of the multiplication ring.
  • Multiplicative commutativity: This is naturally satisfied since a * b = b * a for any real numbers a and b.
  • Multiplicative Identity: The number 1 serves as the identity element.
  • Multiplicative inverse: For every nonzero real number a, there is 1/a such that a * (1/a) = 1.

Here is a visual demonstration of field construction by real numbers:

Add Multiplication to reverse Identification

Relation between rings and fields

Every field is naturally a ring because it satisfies all the axioms that define a ring. However, not every ring is a field. Here are some key differences:

  • In a field, every nonzero element has a multiplicative inverse; this is not required in a ring.
  • Multiplication must be commutative in a field, but this is not necessary in a ring.

Let's look at clear examples to understand these differences:

Example 1: A ring that is not a field

Consider the set of integers (mathbb{Z}). It forms a ring, but not a field. The reason is simple: not every integer has a multiplicative inverse that is also an integer. For example, there is no integer x such that 2 * x = 1.

Example 2: A field that is also a ring

Recall the field of rational numbers (mathbb{Q}). This set satisfies both the ring and field properties, since any rational number a/b (where b ≠ 0) has an inverse b/a which is also a rational number.

Conclusion

The concepts of rings and fields are fundamental to abstract algebra, providing a framework for working with a wide variety of mathematical objects. Rings generalize basic arithmetic operations such as addition and multiplication with applicable systems, while fields add a layer of complexity by enabling division.

These algebraic structures enhance our understanding of number systems and find profound applications in other fields including computer science, cryptography, and coding theory. Mastering these topics opens the door to advanced mathematical studies and a deeper understanding of the underlying principles of mathematics.


Graduate → 2.2


U
username
0%
completed in Graduate

← Prev (2.1.4)
Sylow Theorems
Next (2.2.1) →
Ideals and Factor Rings

Comments 



Rings and Fields

https://www.buddymath.com/g/2.2?bmal=en