Unformatted text preview: Z ŒxŁ is a UFD. In Z ŒxŁ , 1 is a greatest common divisor of 2 and x , but 1 62 2 Z ŒxŁ C x Z ŒxŁ . Lemma 6.6.4. In a unique factorization domain, every irreducible is prime. Proof. Suppose an irreducible p in the unique factorization R divides a product ab . If b is a unit, then p divides a . So we can assume that neither a nor b is a unit. If g 1 ²²² g ` and h 1 ²²² h m are irreducible factorizations of a and b , respectively, then g 1 ²²² g ` h 1 ²²² h m is an irreducible factorization of ab . Since p is an irreducible factor of ab , by Lemma 6.6.1 p is an associate of one of the g i ’s or of one of the h j ’s. Thus p divides a or b . n Corollary 6.6.5. Let R be a unique factorization domain domain. Con-sider the following properties of an nonzero, nonunit element p of R : ³ pR is a maximal ideal. ³ pR is a prime ideal. ³ p is prime. ³ p is irreducible....
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- Fall '08
- Algebra, Prime number, Greatest common divisor, Lemma, Integral domain, Unique factorization domain, Principal ideal domain