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Let F be a filed and S be a subring of F,To prove that S is an integral domain we must show that S is 1- S is commutative ring.2- S not have a zero divisor.Since F is commutative ring then also the subring S is commutativeLet S have a zero divisor, then there exists , and such that ab=0 since F is filed and there exists with , ab=0abb-1=0b-1a(1)=0a=0which is contradiction with Therefore S not have a zero divisor. And it is an integral domain.
 
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Zero divisors
Definition
An element x ≠ 0 in a ring R is a zero divisor in R if xy = 0 for some y ≠ 0 in R
From this definition
Let xy = 0, if x ≠ 0 then x-1 is exists, thus x-1x y = x-10 =0, implies 1y = y = 0, which implies there is no zero divisor
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