## Ring

### Ring

An algebraic structure (R, +, .) consisting of a non-empty set R and two binary operations, called addition (+) and multiplication (.) is called a ring, if following properties are satisfied-

1) Closure Property:
a + b ∈  R, ∀ a , b ∈  R.

2) Associativity Property:
(a + b) + c = a + ( b + c ), ∀ a, b, c ∈  R.

3) Existence of Identity:
a + 0 = a = 0 + a, ∀ a ∈  R.

4) Existence of Inverse:
a + ( -a ) = 0 = ( -a ) + a.

5) Commutative:
a + b = b + a, ∀ a, b ∈  R.

Types of Rings:

1) Null Ring:
The binary operations addition (+) and multiplications (.) defined by ‘0+0 = 0’ and ‘0.0 = 0’ is a ring, called Null Ring or Zero Ring.

2) Commutative Ring:
If the multiplication in Ring is defined by a.b = b.a ∀ a , b ∈  R, than ring is known as Commutative Ring.

3) Ring with Unity:

If an element in Ring is defined by e.a = a.e, ∀  a, b ∈  R, than ring is known as Ring with Unity. Element e is called unit element or identity of R.

#### FIELD

A ring R with at least two elements is called a field if,
1. It is commutative
2. It has unity
3. Each non-zero element possesses multiplicative inverse.

Problems based on above study:

Prob. Prove that a ring R is commutative, if and only if
( a + b )2 = a2 + 2ab + b2 , ∀ a, b ∈   R.

Prob. Show that the polynomial x2 + x + 4 is irreducible over F, the field of integer modulo 11.

Prob. If R is a ring, such that a2 = a, ∀ a ∈  R. Prove that,
1)      a + a = 0, ∀ a ∈  R
2)      a + b = 0 implies a = b ∀ a, b ∈  R
3)      R is a commutative ring.