College Algebra Exam Review 286

College Algebra Exam Review 286 - Proof. Exercises 6.5.1...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
296 6. RINGS (b) For all d , if d divides each a i , then d divides c . Elements a i ;:::;a s in an integral domain are said to be relatively prime if 1 is a gcd of f a i g . Given any Euclidean domain R , we can follow the proofs given for the integers and the polynomials over a field to establish the following results: Theorem 6.5.8. Let R be a Euclidean domain. (a) Two nonzero elements f and g 2 R have a greatest common divisor which is contained in the ideal Rf C Rg . The greatest common divisor is unique up to multiplication by a unit. (b) Two elements f and g 2 R are relatively prime if, and only if, 1 2 Rf C Rg . (c) Every ideal in R is principal. (d) Every irreducible element is prime (e) Every nonzero, nonunit element has a factorization into irre- ducibles, which is unique (up to units and up to order of the factors).
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Proof. Exercises 6.5.1 through 6.5.3 . n This result suggests making the following denitions: Denition 6.5.9. An integral domain R is a principal ideal domain (PID) if every ideal of R is principal. Denition 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 prin-cipal 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

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online