Unformatted text preview: Deﬁnition 6.5.5. A nonzero nonunit element a in an integral domain is said to be prime if whenever a divides a product bc , then a divides one of the elements b or c . An easy induction shows that whenever a prime element a divides a product of several elements b 1 b 2 ³³³ b s , then a divides one of the elements b i . In any integral domain, every prime element is irreducible (Exercise 6.5.19 ), but irreducible elements are not always prime. Deﬁnition 6.5.6. Two elements in an integral domain are said to be associates if each divides the other. In this case, each of the elements is equal to a unit times the other. Deﬁnition 6.5.7. A greatest common divisor (gcd) of several elements a i ;:::;a s in an integral domain R is an element c such that (a) c divides each a i , and...
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 Fall '08
 EVERAGE
 Algebra, Fractions, Complex Numbers, Prime number, Greatest common divisor, Divisor, Integral domain

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