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 ↓


Ideals and Quotient Rings


In abstract algebra, the concepts of ideal and quotient rings are central to understanding the structure of rings. Rings have algebraic structures consisting of a set equipped with two binary operations, usually called addition and multiplication, where the set is closed, associative, and has an additive identity and inverse.

To delve deeper into this area, let's discuss two important sub-structures within rings: ideal and quotient rings. These structures help mathematicians understand how rings can be broken down and analyzed.

Understanding the ideals

An ideal is a special subset of a ring. Let's take a look at the formal definition:

Let ( R ) be a ring, and ( I ) be a subset of ( R ). ( I ) is an ideal of ( R ) if:

  1. ( I ) is a subring of ( R ).
  2. For every ( r in R ) and every ( a in I ), ( ra ) and ( ar ) are both in ( I ).

This means that for (I) to be an ideal it is not enough to be a subring; it must also absorb the multiplication by any element of the ring (R).

Examples of ideals

Example 1: Consider the ring of integers ( mathbb{Z} ). An example of an ideal in ( mathbb{Z} ) is the group of even numbers. Let ( I = 2mathbb{Z} = { ldots, -4, -2, 0, 2, 4, ldots } ).

For any integer ( n ) and even number ( m ), ( nm ) and ( mn ) are both even numbers, which means ( nm, mn in I ). Therefore, ( I ) is a multiple of ( mathbb{Z} ).

even ideal of Z (2Z)
----------------
| ...   | -4   | -2   | 0   | 2   | 4   | ...   |
----------------

Example 2: In the ring ( mathbb{Z}_5 ) (integers modulo 5), the subset ( { 0, 5, 10, 15, ldots } ) forms an ideal. Here ( I ) is simply ( {0} ) since the multiples of 5 are equal to 0 in ( mathbb{Z}_5 ).

Ideals in Z5 : {0}

Types of ideals

There are two main types of ideals:

1. Proper ideals

An ideal ( I ) of a ring ( R ) is called a proper ideal if ( I neq R ). This means that every element of ( R ) is not in ( I ).

2. Dominant ideals

An ideal is called a principal ideal if it can be generated by a single element. If ( a ) is an element of a ring ( R ), then the principal ideal generated by ( a ) is:

(a) = { ra : r in R }

Example: In ( mathbb{Z} ), the ideal ( (3) ) is the set of all multiples of 3, i.e., ( { ldots, -6, -3, 0, 3, 6, ldots } ).

Principal Ideals (3)

Quotient ring

Once we have an ideal, we can build a new structure called a quotient ring. The basic idea is to 'divide' the ring ( R ) by the ideal ( I ).

Formally, the quotient ring, denoted ( R/I ), consists of all cosets of ( I ) in ( R ). Cosets are a way of breaking a ring into disjoint 'blocks' that together form a ring.

Construction of quotient rings

If ( R ) is a ring and ( I ) is a modulus of ( R ), then the set of cosets:

R/I = { r + I : r in R }

The quotient of ( R ) by ( I ) is called a ring. Operations on ( R/I ) are defined by:

  • Sum: ( (a + I) + (b + I) = (a + b) + I )
  • Multiplication: ( (a + I)(b + I) = (ab) + I )

Properties of quotient rings

The quotient ring ( R/I ) has some interesting properties:

  • It is a ring in itself.
  • A ring homomorphism from ( R ) to ( R/I ) maps each element of ( R ) to its corresponding coset in ( R/I ).

Example of a quotient ring

Example: Consider the ring of integers ( mathbb{Z} ) and its ideal ( 3mathbb{Z} ). The quotient ring ( mathbb{Z}/3mathbb{Z} ) is often denoted ( mathbb{Z}_3 ).

The elements of ( mathbb{Z}_3 ) are cosets:

  • ( 0 + 3mathbb{Z} ) (which corresponds to 0)
  • ( 1 + 3mathbb{Z} ) (which corresponds to 1)
  • ( 2 + 3mathbb{Z} ) (which corresponds to 2)
Z/3Z: {0, 1, 2}

Operations are performed modulo 3. For example, adding ( (1 + 3mathbb{Z}) ) to ( (2 + 3mathbb{Z}) ) gives ( (3 + 3mathbb{Z}) ), which is the same coset as ( (0 + 3mathbb{Z}) ).

Visualization of quotient rings

When working with quotient rings like ( mathbb{Z}_n ), it can be helpful to look at their structure. Each element of the quotient ring can be thought of as a looped 'step' in a cyclic diagram.

Example: Visualizing ( mathbb{Z}_4 )

0123

In this visualization, each movement in the counterclockwise direction through a point represents adding 1 to your current position, modulo 4. Thus, this visualization helps us see how the elements move around in a cycle, which is typical of any quotient ring of integers.

Conclusion

Understanding ideals and quotient rings is fundamental to exploring the deep structure of rings in abstract algebra. These concepts provide a way to dissect and analyze rings, making it easier to understand their properties and determine how they interact with other algebraic structures.

By exploring ideals, we understand subsets of rings that work well in operations on rings, and through quotient rings, we are able to construct new types of rings that can offer more simplified analysis.

We also explored various properties and specific examples of ideals and quotient rings along with visual examples to further improve our understanding. With this foundation, one can begin to explore more advanced topics in ring theory and abstract algebra.


Undergraduate → 1.2.9


U
username
0%
completed in Undergraduate

← Prev (1.2.8)
Fields
Next (1.2.10) →
Polynomial Rings

Comments 



Ideals and Quotient Rings

https://www.buddymath.com/ug/1.2.9?bmal=en