College Algebra Exam Review 298

College Algebra Exam Review 298 - a . If b is a proper...

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308 6. RINGS A characteriziation of UFDs We are going to characterize unique factorization domains by two properties. One property, the so–called ascending chain condition for prin- cipal ideals, implies the existence of irreducible factorizations. The other property, that irreducible elements are prime, ensures the essential unique- ness of irreducible factorizations. Definition 6.6.12. We say that a ring R satisfies the ascending chain con- dition for principal ideals if, whenever a 1 R ± a 2 R ± ²²² is an infinite increasing sequence of principal ideals, then there exists an n such that a m R D a n R for all m ³ n . Equivalently, any strictly increasing sequence of principal ideals is of finite length. Lemma 6.6.13. A unique factorization domain satisfies the ascending chain condition for principal ideals. Proof. For any nonzero, nonunit element a 2 R , let m.a/ denote the num- ber of irreducible factors appearing in any irreducible factorization of
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Unformatted text preview: a . If b is a proper factor of a , then m.b/ < m.a/ , by Lemma 6.6.1 . Now if a 1 R a 2 R is a strictly increasing sequence of principal ideals, then for each i , a i C 1 is a proper factor of a i , and, therefore, m.a i C 1 / < m.a i / . If follows that the sequence is nite. n Lemma 6.6.14. If an integral domain R satises the ascending chain con-dition for principal ideals, then every nonzero, nonunit element of R has at least one factorization by irreducibles. Proof. This is exactly what is shown in the proof of Lemma 6.5.17 . n Lemma 6.6.15. If every irreducible element in an integral domain R is prime, then an element of R can have at most one factorization by irre-ducibles, up to permutation of the irreducible factors, and replacing irre-ducible factors by associates....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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