Algebra is the language of modern mathematics, abstract algebra is a common name for the sub area. In 20th century the term |

**Group**: A non empty set G with a binary operation * is called a group if the following axioms are satisfied:

**1)**Closure axiom : For all a, b $\in$ G

a * b $\in$ G

**2)**Associative axioms : For all a, b, c $\in$ G

a * (b * c) = (a * b) * c

**3)**Identity axiom : G contains element 'e' such that $\forall$ a $\in$ G,

a * e = e * a

=a

'e' is called identity element.

**4)**Inverse axiom : For every a $\in$ G, there exists

a$^{-1}$ $\in$ G such that a * a$^{-1}$ = a$^{-1}$ * a = e

a$^{-1}$ is called inverse of a

A group G with respect to the binary operation * is denoted by (G, *)

In a group (G, *) if a * b = b * a for all a, b $\in$ G then G is called an abelian group or commutative group.

**Semi Group**: A non empty set G with binary operation * is called a semigroup if the closure and associative axioms are satisfied

$\forall$ a, b $\in$ G, a * b $\in$ G

and $\forall$ a, b , c $\in$ G, a * ( b * c) = (a * b) * c

**Finite and Infinite groups**: A group (G, *) is called a finite group if G is a finite set and it is called an infinite group if G is an infinite set.

**Order of a group**: In a group (G, *) the number of elements in G is the order of the group. If G contains n elements, we say that order of G is n and write it as O(G) = n . An infinite group is of infinite order.

A group is an algebraic structure with one binary operation. An algebraic structure with two approximately restricted and related binary operations. This algebraic structure is called a

**ring**.

**Ring Definition**: An algebraic structure (R, +, .) consisting of a non empty set R with two binary operations called addition and multiplication, is called a ring if,

**1)**(R, +, .) is an abelian group.

**2)**The binary operation '.' is associative in R

$\forall$ a, b, c $\in$ R, a.(b.c) = (a.b).c

**3)**The binary operation '.' is both left as well as right distributive over the operation '+' in R.

**1)**a.(b + c) = a.b + a.c

**(left distributive law)**

**2)**(b + c).a = b.a + c.a

**(right distributive law)**

In a ring (R, +, .) the identity element w.r.t addition is denoted by 0 and is called the zero of the ring. Thus,

$\forall$ a $\in$ R, a + 0 = 0 + a = a

In a ring (R, +, .) the inverse of an element a w.r.t addition us denoted by -a and is called additive inverse of a in R. Thus

$\forall$ a $\in$ R, a + ( -a) = (-a) + a = 0

$\forall$ a, b $\in$ R, a - b = a + (-b)

**Example :**(Z, +, .) where + and . are usual addition and multiplication in Z is a commutative ring with unity and without zero divisors. It is neither a skew field nor a field.

For any non zero fixed integer k $\neq$ 1, (kZ, +, .) where kZ = {kn| n $\in$ Z} is a commutative ring without unty and without zero divisors.

**Some of the problems based on abstract algebra:**

**Example 1:**Let Z be the set of integers. Show that Z together with the usual addition of integers (+) is an abelian group.

**Solution:**Let a, b, c be any integers.

**1)**$\forall$ a, b, a + b $\in$ Z

Closure axiom is satisfied.

**2)**Abstract Algebra $\forall$ a, b, c $\in$ Z, a + (b + c) = (a + b) + c, since addition of intefers is associative.

Associative axiom is satisfied.

**3)**There exists element 0 $\in$ Z, such that

$\forall$ a $\in$ Z, a + 0 = 0 + a = a. Therefore identity axiom is satisfied.

'0' is identity element

**4)**For every a $\in$ Z, there exists - a $\in$ Z such that a + ( -a) = (- a) + a = 0

Hence the inverse axiom is satisfied.

- a is the inverse of a under addition.

**5)**For all a, b $\in$ Z, a + b = b + a

Commutative axiom is satisfied.

Hence(Z , +) is an abelian group since the number of elements in Z is infinite.

**Example 2:**In a group (G, *) of integers a * b is defined by a * b = a + b - 1.Find the identity element.

**Solution :**Let e be the identity element.

Then if a $\in$ G, a * e = a

a + e - 1 = a

e - 1 = 0

e = 1

**Example 3:**An element a of ring R is said to be idempotent if a$^{2}$ = a. Show that the only idempotent elements of an integral domain are 0 and 1.

**Solution:**Let R be an integral domian.

Let x $\in$ R such that x$^{2}$ = x.

Supposing x $\neq$ 0,

then x$^{2}$ = x

=> x.x = x.1

=> x = 1

**(cancellation law)**

Again, if x $\neq$ 1,

x$^{2}$ = x

x.x - x = 0

x(x - 1) = 0

x = 0 (R is an integral domain)

Hence x$^{2}$ = x in R

=> x = 1 or x = 0