This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Z x is a UFD. In Z x , 1 is a greatest common divisor of 2 and x , but 1 62 2 Z x C x Z x . 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 g ` and h 1 h m are irreducible factorizations of a and b , respectively, then g 1 g ` h 1 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 : pR is a maximal ideal. pR is a prime ideal. p is prime. p is irreducible....
View Full Document
This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08