Abstract algebra is the study of the algebraic structure, which starts from the basic and the simplest of all structures, which is groups, and then these groups are extended to rings and then finally to the fields. Some of the concepts and definitions of abstract algebra are as follows:

**Group:****Definition 1:**** **A non empty set G with a binary composition " * " is called a group if the following conditions are satisfied:

**G.1**. a * b belongs to G for all a, b belonging to G. (Closure Law).

**G.2**. a * (b * c) = (a * b) * c for all a, b, c belonging to G. (Associative Law).

**G.3**. There exists an element e belonging to G, such that

a * e = e * a = a for all a belonging to G. (Existence of identity, where e is called the identity in G).

**G.4**.
For each a belonging G, there exists an element a’ belonging to G such
that a * a’ = a’ * a = e. (Existence of inverse, where a’ is called the
inverse of a and is written as a’ = a$^{-1}$).

We write a group G with respect to binary composition * as (G, *).

**Definition 2:** A group (G, *) is called Abelian or commutative if a * b = b * a for all a, b belonging to G.

**Definition 3:** If
a be an element of a group G, then for any positive integer n, a$^n$ is
defined as a$^n$ = a * a * a * …. (n times) and a$^0$ = e. Also a$^{-1}
= (a$^{-1})^n$.

**Ring:****Definition 4:** A ring is a non empty set R with two binary compositions, denoted by + and * respectively, satisfying the following properties:

**1)**. (R, +) will form an abelian group.

**2)**. a, b belongs to R, for all a, b belonging to R.

**3)**. a * (b * c) = (a * b) * c, for all a, b, c, belonging to R.

**4)**. For all a, b, c belonging to R,

a * (b + c) = a * b + a * c, which is known as the left distributive law.

(b + c) * a = b * a + c * a, which is known as the right distributive law.

If
all the above mentioned 4 properties are satisfied by a set of numbers
belonging to R, then we say, the set is a ring, which is denoted as [R,
+, *]. **Zero Divisor:** A
non zero element a of a commutative ring is called a zero divisor if
there exists some non zero element b in R such that a * b = 0.

A ring [R, +, *] is called a commutative ring if a * b = b * a for all a, b belonging to R.

A
ring [R, +, *] is called a field if it’s non zero elements form an
abelian group with respect to the composition ‘*’ i.e. if R is a field,
then a belonging to R, where a not equal to zero, implies a${-1}$
belongs to R.

**Integral Domain:**A commutative ring is called an integral domain if it has no zero divisors in it.

**The following is the diagrammatic representation of the concepts of abstract algebra:** A group G is called finite or infinite according as it contains a finite or infinite number of elements.

If a group G contains n elements, we say that the order of G is n and write it as o(G) = n.

Based on the above mentioned definitions, some of the examples which come under the concept of abstract algebra are:

Examples of Groups:**1)**. (I, +) is a group, where 0 is the identity and the inverse of a belonging to I is –a belonging to I.

**2)**. Similarly, (R, +), (Q, +) are groups.

**3)**. (N, +) is not a group, as the axioms G.3. and G.4. are not satisfied in N.

**4)**. The set I of integers is not a group under usual multiplication, since the axiom G.4. is not satisfied in I.

**5)**.
The set R* of all non- zero real numbers is an abelian group under
usual multiplication, where I is the identity and the inverse of a
belonging to R* is 1/a.

**6)**. (I, +), (R, +), (Q*, .), (R*, .) are infinite abelian groups.

**Some of the examples of Rings are:****1)**. The set E of all even integers is a commutative ring without unit element.

**2)**. The set of all 2 by 2 matrices as shown below:

M_{2}(R) = $\begin{Bmatrix}

\begin{pmatrix}

a & b\\

c& d

\end{pmatrix}|a,b,c\ and\ d\ belongs\ to R

\end{Bmatrix}$

is a non commutative ring under the addition and multiplication of 2 by 2 matrices.

**3)**.
The set of integers with two binary compositions * and o defined by a *
b = a + b – 1, aob = a + b – ab for all integers a and b is a
commutative ring with unity.

**The following diagram represents the types of Rings:** **The following are some examples of integral domain and fields:****1)**. The ring of integers I is an integral domain because for any tow integers a and b: a . b = 0 implies a = 0 or b = 0.

**2)**. Q and R are fields with respect to the usual addition and multiplication.

**3)**. For any prime p, Jp = {0, 1, 2, …., p - 1} is a field with respect to addition and multiplication modulo p.

**4)**. The set C = {a + bi: a, b belonging to R} of complex numbers is a
field under the usual addition and multiplication of complex numbers.