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Abstract Algebra

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Algebra is the language of modern mathematics, abstract algebra is a common name for the sub area. In 20th century the term abstract algebra was coined to distinguish from other parts of algebra. Abstract algebra introduces students to that language through a study of solving inequalities, linear functions, quadratic functions, polynomials, exponents, groups, matrices, rings, radical expressions, rational expressions, etc.

In algebra, students study about mathematical quantities. A mathematical quantity is anything that can be measured or counted. Some quantities remain constant. Other changes or varies and are called variable quantities. Generally quantities are used to form expressions, equations and inequalities.

Variables are used in place of numbers in equations so that it allows the statement of relationships among numbers that are unknown or unspecified. Abstract algebra is basically the study of abstract part of algebra, such as : sets, groups, rings and many more. It is also known as modern algebra. Let us learn what abstract algebra is all about.

Topics

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Abstract algebra is an advanced set of topics related to algebra, including rings, fields, groups, ideals etc. Some examples based on above topics are explained below.
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.

Abstract Algebra Problems

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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