1. Ring Theory

2. Introduction to ring theory Definition(Ring):- A ring R is a set with two binary operation addition and multiplication such a that for all a,b,c in R . 1) a+b∈R. 2) a+b=b+a. 3) (a+b)+c=a+(b+c). 4) There is an element 0 in R such that a+0=a. 5)There is an element –a in R such that, a+(-a) =0.

3. 6)a.b∈R
7)a.(b.c)=(a.b).c
8) a.(b+c)=a.b+a.c and (b+c).a=b.a+c.a
Examples:-
The sets of integers (z),rational numberes(Q), Real numbers(R),Complex numbers(c).
2) Let n∈N. The set of all n×n matrices over R.
3) The set of even integers.

4. Subrings Definition:- A Subset S of a ring R is a subring of R if S is itself a ring with the operation of R. Example:- 1){0} and R are subrings of any ring R. 2){0,2,4} is a subring of the ring Z6.

5. C R
Z[i]= {a+bi / a,b ∈Z } Q(√2)= {a+ b√2 / a,b ∈Q } Q Z
5Z Z Z Z
10Z Z Z Z
8Z Z Z