Copyright

Rings: Binary Structures & Ring Homomorphism

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

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

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

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:03 Rings
  • 1:02 Binary Structures
  • 1:27 Ring Rules
  • 2:44 Ring Homomorphism
  • 3:19 Some Examples
  • 4:23 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed Speed
Lesson Transcript
Instructor: Russell Frith
In this lesson, basic definitions are presented for algebraic rings and ring homomorphisms. These form the foundations for a broader discourse in abstract algebra. Rings are used in more complex algebraic structures and have many applications in computer science, physics, and engineering.

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.

Binary Structures

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


null


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, b, and c be elements of 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 S is defined as follows:


ring homomorphism


Some Examples

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

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

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support