integral domain

INTEGRAL DOMAIN
Definition 1 - 16
An integral domain is a commutative ring with identity which has no
zero divisors ( a commutative ring with identity R is called integral domain
if xy = 0 , implies x = 0 or y = o , for x , y ? R .

Example
Z and Z5 are an integral domain , but Z4 is not .

Exercise
Let R be a ring with more than one element . If ax = b has a solution
for all nonzero a? R and with for all b ? R , then R is an integral domain.
Solution
Suppose that a ? 0 , b ? 0 , and ab = 0 ,
then , abx = 0 for all x ? R .
So ? c ? R , bx = c has a solution ?
Thus , ac = 0 .
Because , for some c ? R , ac = a , we get a = 0 , a contradiction .
R must be an integral domain .

Exercises
1 - Prove that Zp is an integral domain .

2 – Is any subring with identity of integral domain is integral domain ?
Prove your answer .
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