Factoring Ideals in Integral Domains by Marco Fontana Evan Houston & Thomas Lucas

Author:Marco Fontana, Evan Houston & Thomas Lucas
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

We record the following simple consequence of Theorem 4.2.12 for ease of reference in Example 4.3.4 below.

Corollary 4.2.13.

Let R be a Prüfer domain with finite unsteady character. If each steady maximal ideal is invertible and each nonzero nonmaximal prime is sharp and contained in a unique maximal ideal, then R has weak factorization.

One of the main concepts studied by Loper and Lucas in 2003 [58] is how far an almost Dedekind domain is from being Dedekind. For example, from what has already been observed in Sect. 3.2, an almost Dedekind domain R has sharp degree 2 if it is not Dedekind, but the intersection is Dedekind, where the intersection is taken over all maximal ideals M that are not invertible in R (in this case, they coincide with the dull maximal ideals of R, considered in Sect. 3.2). Note that R must have infinitely many invertible maximal ideals for this to happen. An example in [19] shows that an almost Dedekind domain with infinitely many noninvertible maximal ideals can have sharp degree 2 [19, Example 3.2]. Higher sharp degrees (including infinite ordinal degrees) can be defined recursively, as in Sect. 3.2. For example, an almost Dedekind domain R has sharp degree 3, if R2 is not Dedekind and with R3 Dedekind where R3 is the intersection of the localizations at the (nonempty set of) noninvertible maximal ideals of R2. It turns out that an almost Dedekind domain that is not Dedekind has weak factorization if and only if it has sharp degree 2. More precisely, Theorem 4.1.5 essentially shows that an almost Dedekind domain with sharp degree 2 has weak factorization. In Proposition 4.2.14 below, we establish the converse, as well other ways to detect weak factorization in almost Dedekind domains.

Proposition 4.2.14.

Let R be an almost Dedekind domain that is not Dedekind. The following are equivalent. (i) R has weak factorization.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.