Home » Discrete Mathematics

Rings and Types of Rings | Discrete Mathematics

In this article, we will learn about the introduction of rings and the types of rings in discrete mathematics.
Submitted by Prerana Jain, on August 19, 2018

Ring

The algebraic structure (R, +, .) which consisting of a non-empty set R along with two binary operations like addition(+) and multiplication(.) then it is called a ring.

An algebraic ( or mathematically) system (R, *, o) consisting of a non-empty set R any two binary operations * and o defined on R such that:

  1. (R, *) is an abelian group and (R, 0) is a semigroup.
  2. The operation o is the distributive over the operation * is said to be the ring.

There are following postulates are satisfied:

R1

The system (R, +) is an abelian group. So we have the following properties:

1. Closure property

The set R is called with respect to the composition +.

i.e. aER, bER => a+b E R for all a, b E R

2. Associativity

Associative law hld good in the set R for the composition +.

i.e. (a+b) + c = a + (b+c) for all a, b, c E R.

3. Existence of identity

There exist an unique 0 E R called zero element such that, a + b = a = 0 + a, a E R

4. Existence of inverse

For each a E R there exists an elements a E R such that, a + (-a) = 0 = (-a) + a

5. Commutative of addition

Commutative law holds good in the set R for the composition +.

i.e. a + b = b + a for all a, b E R

R2

The set R is closed with respect to the multiplication composition.

R3

Multiplication composition is associative i.e. (a.b).c = a.(b.c) for all a, b, c E R

R4

The multiplication composition is right and left distributive with respect to addition.

i.e. a.(b + c) = a.b + a.c for every a, b, c E R (left distributive law)

(b + c). a = b.a + c.a (right distributive law)

Types of Rings

1. Null Ring

The singleton (0) with binary operation + and defined by 0 + 0 = 0 and 0.0 = 0 is a ring called the zero ring or null ring.

2. Commutative Ring

If the multiplication in a ring is also commutative then the ring is known as commutative ring i.e. the ring (R, +, .) is a commutative ring provided.

a.b = b.a for all a, b E R

If the multiplication is not commutative it is called non- commutative ring.

3. Ring with unity

If e be an element of a ring R such that e.a = a.e = a for all E R then the ring is called ring with unity and the elements e is said to be units elements or unity or identity of R.

4. Ring with zero divisor

A ring (R, +, .) is a said to have divisor of zero (or zero divisor), if there exist two non-zero elements a, b E R such that a.b = 0 or b.a = 0 where 0 is the additive identity in R . here a and b are called the proper divisor of zero.

5. Ring without zero divisor

A ring R is said to be without zero divisor. If the product of no two non zero elements of R is zero i.e. if ab = 0 => a = 0 or b = 0.

Properties of Ring

If R is a ring then for all a, b, c E R

  1. a0 = 0a = 0
  2. a(-b) = -(ab) = (-a)b
  3. (-a)(-b) = ab
  4. a(b-c) = ab – ac
  5. (b-c)a = ba- ca




Comments and Discussions

Ad: Are you a blogger? Join our Blogging forum.



Languages: » C » C++ » C++ STL » Java » Data Structure » C#.Net » Android » Kotlin » SQL
Web Technologies: » PHP » Python » JavaScript » CSS » Ajax » Node.js » Web programming/HTML
Solved programs: » C » C++ » DS » Java » C#
Aptitude que. & ans.: » C » C++ » Java » DBMS
Interview que. & ans.: » C » Embedded C » Java » SEO » HR
CS Subjects: » CS Basics » O.S. » Networks » DBMS » Embedded Systems » Cloud Computing » Machine learning » CS Organizations » Linux » DOS
More: » Articles » Puzzles » News/Updates

© https://www.includehelp.com some rights reserved.