Page 52 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Euclidean Rings

An integral domain R is said to be a Euclidean ring if for every a 0 in R there is defined a non-
negative integer, to be denoted by d (a), such that:

a
(i) for all ,a b R , both non-zero, ( )d a d ( ) ,
b
(ii) for any ,a b R , both non-zero, there exists ,q r R such that a q b r when either
b
r = 0 or ( )d r d ( ).
Illustrative Examples

Example 26: Prove that the ring of complex numbers C is an integral domain.
a
Solution: Let J (i) = {a bi : ,b } I .
It is easy to prove that J (i) is a commutative ring with unity.
The zero element 0 + 0.i and unit element 1 + 0.i.
Also this ring is free from zero-divisors because the product of two non-zero complex numbers
cannot be zero. Hence J (i) is an integral domain.

Example 27: Prove that set of numbers of the form a b 2 with a and b as integers is an
integral domain with respect to ordinary addition and multiplication.

a
Solution: Let D {a b 2 : ,b I }
I
( )( , ) is an abelian group.
D
1
b
,
(I ) Let a b 2 D and a b 2 D , then a , , a b I
11 1 1 2 2 1 1 2 2
Now, (a b 2) (a b 2) (a a ) (b b ) 2 D
1 1 2 2 1 2 1 2
as a 1 a 2 , b 1 b 2 I .
Hence D is closed under addition.
(I ) Addition is associative in the set of real numbers.
12
(I ) 0 (0 0 2) D is the additive identity in D because 0 I.
13
(I ) If (a b 2) D
14
Then ( a ) ( b ) 2 D and (a b 2) [( a ) ( b ) 2] 0 0 2 0 the additive identity. Hence
each element in D possesses additive inverse.
(I ) Addition is commutative in the set of real numbers.
15
I (D, .) is semi-abelian group with unity.
2
,
(I )(a b 2)(a b 2) (a a 2b b ) (a b a b ) 2 D as a b 2b b a b I for
21 1 2 2 2 1 2 1 2 1 2 2 1 1 2 1 2 1 1
,
a 1 , b a 2 ,b 2 I .
1

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