Unformatted text preview: Proof. Exercises 6.5.1 through 6.5.3 . n This result suggests making the following deﬁnitions: Deﬁnition 6.5.9. An integral domain R is a principal ideal domain (PID) if every ideal of R is principal. Deﬁnition 6.5.10. An integral domain is a unique factorization domain (UFD) if every nonzero element has a factorization by irreducibles that is unique up to order and multiplication by units. According to Theorem 6.5.8 , every Euclidean domain is both a principal ideal domain and a unique factorization domain. The rest of this section will be devoted to a further study of principal ideal domains and unique factorization domains; the main result will be that every principal...
View
Full Document
 Fall '08
 EVERAGE
 Algebra, Integers, Greatest common divisor, Integral domain, Unique factorization domain, Euclidean domain, Principal ideal domain

Click to edit the document details