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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. By Prerana Jain Last updated : May 09, 2023

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 + 0 = 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

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