Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Algebra → Abstract Algebra ↓


Rings


Abstract algebra is a field of mathematics that studies algebraic structures such as groups, rings, and fields. In this article, we will take a deep look at one of these important structures called rings. Rings are important to understand because they are fundamental concepts that lead to many other areas of mathematics and have applications in science and engineering.

What is a ring?

A ring is a set equipped with two binary operations that we can think of as addition and multiplication. These operations must satisfy certain properties that generalize the arithmetic we are familiar with. The formal definition is as follows:

A ring (R) is a set equipped with two binary operations (+) (called addition) and (cdot) (called multiplication) satisfying the following axioms:

  • (R,+) is an abelian group which means:
    • Closure: for all (a, b in R), (a + b in R).
    • Associativity: For all (a, b, c in R), ((a + b) + c = a + (b + c)).
    • Identity element: There exists an element (0 in R) such that, for all (a in R), (a + 0 = a).
    • Inverse Element: For every (a in R), there exists (-a in R) such that (a + (-a) = 0).
    • Commutativity: For all (a, b in R), (a + b = b + a).
  • (R, (cdot)) is a semigroup which means:
    • Closure: For all (a, b in R), (a cdot b in R).
    • Associativity: For all (a, b, c in R), ((a cdot b) cdot c = a cdot (b cdot c)).
  • Distributive Law: Multiplication is distributive over addition.
    • For all (a, b, c in R), (a cdot (b + c) = a cdot b + a cdot c).
    • For all (a, b, c in R), ((a + b) cdot c = a cdot c + b cdot c).

Examples of rings

1. Set of integers:

The set of integers (mathbb{Z}) forms a ring with the usual operations of addition and multiplication. Let's see why:

  • Closure: Adding or multiplying any two integers will always give another integer.
  • Associativity: Addition and multiplication of integers are both associative.
  • Additive identity: The integer 0 serves as the identity for addition.
  • Additive Inverse: For every integer (a) there is an integer (-a) such that (a + (-a) = 0).
  • Commutativity: Integer addition is commutative, i.e., (a + b = b + a).
  • Distributive Law: Multiplication distributes over addition, i.e., (a(b + c) = ab + ac).
0 1 -1

2. Matrix rings:

Consider the set of all (n times n) matrices having real numbers as entries. This set forms a ring under matrix addition and multiplication. Let us verify this:

  • Closure: The sum or product of any two (n times n) matrices is another (n times n) matrix.
  • Associativity: Matrix addition and multiplication are both associative.
  • Additive identity: The zero matrix, where all entries are zero, serves as the additive identity.
  • Additive Inverse: For any matrix (A), its inverse (-A) is obtained by taking the negative of each entry in (A), giving (A + (-A)) the zero matrix.
  • Distributive Law: Matrix multiplication distributes over matrix addition.

Properties and types of rings

1. Interchangeable rings:

A ring is commutative if its multiplication is commutative, i.e. for any (a, b in R), (a cdot b = b cdot a). For example, the ring of integers (mathbb{Z}) is commutative.

2. Rings with unity:

A unit ring (or identity ring) has an element (1 in R) such that for any (a in R), (a cdot 1 = a) and (1 cdot a = a). The set of integers (mathbb{Z}) is a unit ring.

3. Zero divisor:

In a ring, a nonzero element (a) is called a zero divisor if there exists a nonzero element (b) such that (a cdot b = 0). For example, in the ring of (2 times 2) matrices, a has zero divisors.

4. Integral domain:

An integral domain is a commutative ring that contains unity and has no zero divisors. The ring of integers (mathbb{Z}) is an example of an integral domain.

5. Division ring:

A division ring (or skew field) is a ring in which every nonzero element has a multiplicative inverse. Note that multiplication in a division ring need not be commutative. The set of nonzero real numbers under multiplication forms a division ring.

Viewing ring operation

Let's look at some basic operations within the ring using a simple number line.

-2 -1 0 1 2 3 4 5

Here, (0) represents the additive identity (green dot), and (2) represents a random positive element in our ring (red dot).

Sum example:

If we add (2) and (3), we move to the right on the number line starting from (2). So,

2 + 3 = 5

Here, (5) is another element in our ring, represented at the rightmost point.

Multiplication examples:

Considering multiplication, if we choose to multiply (2) and (3), we repeat the process of 'adding' (2) a total of three times:

2 cdot 3 = 6

The product (6) if extended is another point on the number line remaining within the integer set.

Understanding ideal types

1. Left ideal:

A subset (I) of a ring (R) is called a left ideal if it satisfies the following conditions:

  • ((I, +)) is a subgroup of ((R, +)).
  • For every (r in R) and every (x in I), (r cdot x in I).

2. Right ideal:

Similarly, (I) is a perfect ideal if:

  • ((I, +)) is a subgroup of ((R, +)).
  • For every (r in R) and every (x in I), (x cdot r in I).

3. Two-way model:

A subset (I) is a bipartite ideal if it is both a left ideal and a right ideal. Therefore, it satisfies:

  • ((I, +)) is a subgroup of ((R, +)).
  • For every (r in R) and every (x in I), (r cdot x in I) and (x cdot r in I).

Conclusion

In this introduction to rings, we have discussed the definitions, properties, and some examples of rings in abstract algebra. Rings are important in a variety of areas within and outside mathematics. From the integers that form the basis of our understanding of numbers to matrix rings that model complex systems, their applications are numerous.

By understanding the fundamentals of rings, you can equip yourself with the tools to explore more advanced topics such as modules, fields, and beyond.


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