### 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.
Solution:

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

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.
Solution:

Prob. Define field and prove that the set F = { 0, 1, 2, ….,6 } under addition and multiplication modulo 7 is a field.

Solution:

Prob. Define field. Prove that the set {0, 1, 2} (mod 3) is a field with respect to addition and multiplication (mod 3).

Solution: