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 ´ ¤ ²²² 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 ﬁnite. n Lemma 6.6.14. If an integral domain R satisﬁes 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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- Fall '08
- Algebra, Integral domain, Ring theory, Commutative ring, Unique factorization domain, principal ideals