Page 43 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

Therefore, the set of Gaussian integers is a commutative ring with unity as multiplicative Notes
identity.

Example 14: Prove that the set of all real numbers of the form m n 2 where m, n are
rational numbers is a ring under the usual addition and multiplication.

Solution: Let R = { m n 2 : m, n are real numbers}.

R (R, +) is abelian group.
1
(R ) Let , m n 2 , m n 2 R then
11 1 1 2 2

(m 1 n 1 2) (m 2 n 2 2) (m 1 m 2 ) (n 1 n 2 ) 2 R
because sum of two real numbers is a real number.

(R 12 )(m 1 n 1 2) (m 2 n 2 2) (m 1 n 2 2)(m 1 n 2 2)(m 1 n 2 2)

because addition of real numbers is a real number
(R ) Associative law for addition of real numbers holds, i.e.,
13
(m 1 n 1 2) {(m 2 n 2 2) (m 3 m 3 2)}

{(m 1 n 1 2 (m 2 n 2 2)} (m 3 m 3 2)
for m ,n ,m ,n ,m ,n 3 to be rational numbers.
n 1 2 2 3
(R ) 0 ( 0 0. 2) Î R is the identity of addition in R.
14
(R ) Let m n 2 R , then – (m n 2)
15
= m n 2 R and also

(m n 2) ( m n 2) (m n ) (n n ) 2 0

Hence additive inverse for each element in R exists in R.
R (R, .) is a semi-group.
2
(R ) (m n 2) (m n 2)
21 1 1 2 2
= (m m 2 2n n 2 ) (m n 2 m n 1 ) 2
1
1
2
1
= a b 2 R
as a and b being the sums of products of rational numbers are rational.
(R ) Multiplication is associative in R, i.e.,
22
(m n 2)(m n 2) (m n 2)
1 1 2 2 3 3
(m n 2)(m n 2) (m n 2)
1 1 2 2 3 3
R Multiplication is left as well as right distributive over addition in R. Hence R is a ring
3
under usual addition and multiplication.

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