*Russell Frith*

# Rings: Binary Structures & Ring Homomorphism

## Rings

Let's say you are in Honolulu and you want to call your friend in Tokyo. Your local time is 11:45 AM and you know Tokyo is 19 hours ahead of you. If you call your friend, will she be awake or not?(11:45 AM plus 19 hours makes the time in Tokyo to be 6:45 AM. When you add 19 hours, you're performing an arithmetic calculation known as clock arithmetic.

This type of addition is explained algebraically using the notion of **rings.** Let *S* be a set which is a container holding an unspecified number of named objects designated as *a,* *b,* *c,* and so on. The elements of the set may be subjected to **set operations** such as addition and multiplication. When the elements of a set are transformed by some set operation, the result of the transformation is also an existing element within the set. If the result of the set operation generates a result which is not in the given set, then the set is not a ring.

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

## You must cCreate an account to continue watching

### Register to view this lesson

As a member, you'll also get unlimited access to over 84,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Get unlimited access to over 84,000 lessons.

Try it now*It only takes a few minutes to setup and you can cancel any time.*

###### Already registered? Log in here for access

Back*Coming up next:*How to Write Sets Using Set Builder Notation

### You're on a roll. Keep up the good work!

### Just checking in. Are you still watching?

Yes! Keep playing.## Binary Structures

A binary operation on a set *S*, designated by the symbol *#,* is simply a function that's formulated as a small function:

In other words, given any two elements *s1* and *s2* from *S,* there is a well-defined element *s1 # s2* in *S.* An algebraic set that exhibits this feature without any exception is known as a binary structure.

## Ring Rules

A **ring** is a set *R* endowed with two binary operations + and *, called **addition** and **multiplication**, respectively, which are required to satisfy a rather long list of conditions. Let ** a**,

**, and**

*b***be elements of**

*c**R*. If

*R*is a ring then the following axioms apply:

(A1) **Commutativity of addition**: *a* + *b* = *b* + *a*

(A2) **Associativity of addition**: (*a* + *b*) + *c* = *a* + (*b* + *c*)

(A3) **Additive identity for addition**: There exists an element in *R* called *0* such that 0 + *a* = *a*

(A4) **Additive inverse for addition**: For any element *x* in *R*, there exists an element *y* in *R* such that *x* + *y* = 0

(M1) **Associativity of multiplication**: *a* * (*b* * *c*) = (*a* * *b*) * *c*

(M2) **Multiplicative identity**: There exists an element in *R* called *1* such that 1 * *a* = *a*

(D) **Distributive law**: *a* * (*b* + *c*) = *a* * *b* + *a* * *c* and (*a* + *b*) * *c* = *a* * *c* + *b* * *c*

## Ring Homomorphism

A **homomorphism** between two algebraic objects is a map *f* between two sets which preserves a particular algebraic structure. A **ring homomorphism** between two sets ** R** and

**is defined as follows:**

*S*

## Some Examples

Let's now take a look at some examples of ring-shaped binary structures and ring homomorphism.

### 1. Modulo

A famous example of a ring homomorphism is the **modulo-n** function. This function takes the remainder when an integer is divided by *n*:

For instance, if *f* is the modulo-7 function then 36%7 = 1.

Here is an example of a modulo-n set. A modulo-n set contains the elements {0, 1, 2, ... n-1}.

### 2. Matrices

The set of 2-by-2 matrices with real elements is written in this equation:

This set satisfies the above ring axioms because of the operations of matrix addition and matrix multiplication. The element

is the multiplicative identity for the ring. The following matrix multiplication example shows that 2-by-2 matrix multiplication isn't commutative.

Nevertheless, this set may still be considered a ring.

## Lesson Summary

Let's take a few moments to recap the important information we learned in this lesson.

**Rings** are sets which are closed under addition and multiplication. That is, when any two elements of a set are either added or multiplied, then the result is also in the set of those elements.

Rings generally support:

**Commutativity**,*a*+*b*=*b*+*a***Associativity**, (*a*+*b*) +*c*=*a*+ (*b*+*c*)**Additive inverse**, in which any element*x*in*R*, there exists an element*y*in*R*such that*x*+*y*= 0**Additive identity for addition**, in which there exists an element in*R*called*0*such that 0 +*a*=*a***Multiplicative inverse**, in which there exists an element in*R*called*1*such that 1 **a*=*a*

A **homomorphism** between two algebraic objects is a map *f* between two sets which preserves a particular algebraic structure, while a **ring homomorphism** is a function *f* on a commutative ring that has the following two properties:

*f*(*a*+*b*) =*f*(*a*) +*f*(*b*) and*f*(*ab*) =*f*(*a*) **f*(*b*)

To unlock this lesson you must be a Study.com Member.

Create your account

### Register to view this lesson

### Unlock Your Education

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.

Become a MemberAlready a member? Log In

Back*Rings: Binary Structures & Ring Homomorphism*

Related Study Materials

- Math Courses
- Standardized Tests Courses
- Test Prep Courses
- Study Courses
- ACT and SAT Prep Courses
- High School Algebra I: Help and Review
- High School Algebra I: Tutoring Solution
- Glencoe Geometry: Online Textbook Help
- Calculus: Tutoring Solution
- Quantitative Analysis
- Holt Geometry: Online Textbook Help
- UExcel Precalculus Algebra: Study Guide & Test Prep
- Precalculus: Help and Review
- High School Precalculus: Help and Review
- Math 103: Precalculus
- GED Math: Quantitative, Arithmetic & Algebraic Problem Solving
- GED Science: Life, Physical and Chemical
- GED Social Studies: Civics & Government, US History, Economics, Geography & World

##### Browse by Courses

- The Classification, Genetics & Evolution of Organisms
- Even Number Pairing & Equations
- Explanation or Cause: Definitions & Examples
- Sufficient & Consistent Information in a Text
- Using Patterns to Solve Math Problems
- Quiz & Worksheet - Finding Unnecessary Sentences in Passages
- Quiz & Worksheet - Maintaining Point of View in Essays
- Quiz & Worksheet - Calculating Total Equity
- Quiz & Worksheet - Types of Writing Styles
- Quiz & Worksheet - Structural Unemployment
- Algebraic Word Problems
- Fundamental Algebraic Expressions
- Writing & Solving Multi-Step Equations
- Basic Geometry Calculations
- Accuplacer Math: Quantitative Reasoning, Algebra, and Statistics Placement Test Flashcards

##### Browse by Lessons

- Biology 202L: Anatomy & Physiology II with Lab
- Biology 201L: Anatomy & Physiology I with Lab
- California Sexual Harassment Refresher Course: Supervisors
- California Sexual Harassment Refresher Course: Employees
- Sociology 110: Cultural Studies & Diversity in the U.S.
- Overview of the Sun-Moon-Earth System
- Earth's History and Composition
- Nature of Scientific Investigations
- Overview of Living Organisms
- Force and Motion
- Addressing Cultural Diversity in Distance Learning
- New Hampshire Homeschooling Laws
- Setting Student Expectations for Distance Learning
- COVID-19 Education Trends that are Here to Stay
- What to Do with a COVID-19 College Gap Year
- Active Learning Strategies for the Online Classroom
- How to Promote Online Safety for Students in Online Learning

##### Latest Courses

- Tanka Poems: Lesson for Kids
- Platypus: Habitat & Adaptations
- Songs that Relate to Fahrenheit 451
- Oak Tree Adaptations: Lesson for Kids
- Autocratic Approach to Classroom Management
- Physical Models: Scale Models & Life-Size Models
- What Is Gifted Education? - History, Models & Issues
- Quiz & Worksheet - Social Decision Schemes
- Quiz & Worksheet - Ethical Obligations of Employees to an Organization
- Quiz & Worksheet - Radioactive Nuclei & Decay
- Quiz & Worksheet - Currency Appreciation & Unemployment
- Flashcards - Real Estate Marketing Basics
- Flashcards - Promotional Marketing in Real Estate
- Common Core English & Reading Worksheets & Printables
- Noun Worksheets

##### Latest Lessons

- Animal Science Study Guide
- Middle School US History: Homeschool Curriculum
- Fundamentals of Nursing Syllabus Resource & Lesson Plans
- Intermediate Excel Training: Help & Tutorials
- Biology 101 Syllabus Resource & Lesson Plans
- SHSAT Math: Basic Algebraic Expressions
- History Alive America's Past Chapter 4: How & Why Europeans Came to the New World
- Quiz & Worksheet - The Difference Between Inertia and Mass
- Quiz & Worksheet - Accuracy vs. Precision in Chemistry
- Quiz & Worksheet - The 3-Step Writing Process for Workplace Communication
- Quiz & Worksheet - Characteristics & Importance of Reptiles
- Quiz & Worksheet - Symbiotic Interactions in Disease

##### Popular Courses

- Barriers to Effective Communication: Definition & Examples
- Transforming Quadratic Functions
- Engineering Internships for High School Students
- Study.com's College Accelerator
- Alcohol Awareness Activities
- Summer Tutoring Ideas
- Illinois Common Core Social Studies Standards
- Sequencing Activities for Preschoolers
- How Long Does it Take to Learn Spanish?
- New Jersey Common Core State Standards
- Florida Teacher Certification Renewal
- How to Learn English Quickly

##### Popular Lessons

##### Math

##### Social Sciences

##### Science

##### Business

##### Humanities

##### Education

##### History

##### Art and Design

##### Tech and Engineering

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

##### Health and Medicine

- Find an example of a group G where the elements of G that satisfy the equation x^2 = e do not form a subgroup of G.
- How many units are in M_2(\mathbb{Z}_3), the ring of all 2 \times 2 matrices with entries in \mathbb{Z}_3? Justify your answer?
- Find all the units for each of the following rings. Justify your answers briefly. a. \mathbb{Z}_{15} b. \mathbb{Z}_{11} c. \mathbb{Z} \times \mathbb{Q}\times \mathbb{Z}_3
- Why every subgroup, z_n, under addition is also a subring of z_n?
- Find an example of two prime ideals I,J\in R such that I J is not prime.
- In Example 4.1, assume that the coefficient of friction: In a ring compression test, a specimen 10 mm in , height with outside diameter of 30 mm and inside diameter of 15 mm is reduced in thickness b
- Let F be a field and let G = F times F. Define operations of addition and multiplication on G by setting (a, b) + (c, d) = (a + c, b + d) and (a, b) . (c, d) = (a c, b d). Do these operations define t
- Let \phi be a map from \mathbb{Z}_{12} to \mathbb{Z}_8 defined by \phi (x) - 3x(mod8). Determine if \phi is well-defined, homomorphism, epimorphism, monomorphism, and isomorphism.
- Let R=\mathbb{Z}_8[x] be the polynomial ring over \mathbb{Z}_8=\{\bar{0},\dots,\bar{7}\} on the variable x. Consider f=f(x)=\bar{1}+\bar{2}x+\bar{3}x^2\text{ and } g=g(x)=\bar{4}+\bar{5}x\text{ in }\m
- Let H be a set of all linear combinations h = a + bi + cj + dk. where a. b. and d are real numbers, and i, j, and k are the elements from the group of quaternions. So defined H is a four-dimensional

#### Explore our library of over 84,000 lessons

- Create a Goal
- Create custom courses
- Get your questions answered

**Premium**to add all these features to your account!

**Premium**to add all these features to your account!