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Euclidean-like Characterizations of Dedekind, Krull, and Factorial Domains

✍ Scribed by C.S. Queen

Publisher: Elsevier Science
Year: 1994
Tongue: English
Weight: 473 KB
Volume: 47
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.1994.1042

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✦ Synopsis

Though Euclidean domains are principal ideal domains, the converse is known to be false. We develop a notion like that of the Euclidean ring for which the converse is true. We similarly give new characterizations of Dedekind, Krull, and unique factorization domains. We also introduce the idea of inductive ideal classes and prove results analogous to those obtained by Lenstra for Euclidean ideal classes. 1994 Academic Press. Inc.

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Given a subset X of a Dedekind domain D, and a polynomial F # D[x], the fixed divisor d(X, F) of F over X is defined to be the ideal in D generated by the elements F(a), a # X. In this paper we derive a simple expression for d(X, F) explicitly in terms of the coefficients of F, using a generalized n