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If
are elements of ring such that , then and are called divisors of - (Fraleigh 19.3) In the ring
the divisors of are precisely those non-zero elements that are not relatively prime to . - (Fraleigh 19.4) If
is prime, then has no divisors of . - For any ring, the set of elements that are not divisors of
is closed under multiplication.
- (Fraleigh 19.3) In the ring
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An integral domain
is a Commutative Ring with unity and containing no divisors of . -
(Fraleigh 19.5) The following cancellation laws hold in ring
if and only if is an integral domain -
The direct product of two non-zero rings is not an integral domain.
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A subdomain of integral domain
is a subset of the domain that is an integral domain under induced operations from the whole integral domain. - (Fraleigh e19.24) The intersection of subdomains of
is also a subdomain . - (Fraleigh e19.27) The characteristic of a subdomain of an integral domain
is equal to the characteristic of . - (Fraleigh e19.28)
. Furthermore, this subdomain is contained in every subdomain of . - (Fraleigh e19.29)
or , where is prime.
- (Fraleigh e19.24) The intersection of subdomains of
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(Fraleigh 19.11; Dummit 7.1.3) Every finite integral domain is a field
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Proof: Let
be an integral domain whose elements are . All elements in
of the form are distinct because of the cancellation laws. Also, none of these elements is . Therefore, we can map these to (i.e., for some ).
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Every integral domain is contained in a field called the Quotient Field of the integral domain