Normsets of almost Dedekind domains and atomicity

Abstract

In this paper, we will introduce a new norm map on almost Dedekind domains. We compare and contrast our new norm map to the traditional Dedekind-Hasse norm. We prove that factoring in an almost Dedekind domain is in one-to-one correspondence to factoring in the new normset, improving upon this results in [1]. In [4], an atomic almost Dedekind domain was constructed with a trivial Jacobson radical. We pursue atomicity in almost Dedekind domains with nonzero Jacobson radicals, showing the usefulness of the new norm we introduced. We state theorems with regard to specific classes of almost Dedekind domains. We provide a necessary condition for an almost Dedekind domain with nonzero Jacobson radical to be atomic.

Boundness in almost Dedekind domains

“…The previous results only deal with finitely generated ideals; however, it would be interesting to know if we can also extend those results to principal ideals. An almost Dedekind domain is said to be [4]:…”

“…However, the methods used in [8] were not enough to cover all almost Dedekind domains, because they do not give any result when the set Crit(D) of critical ideals of D coincides with the maximal space of D. Yet, no example of this phenomenon was known; for example, an open question in [4] was if there exists an almost Dedekind domain that is completely unbounded, i.e., where the ideal function associated to every element is unbounded (see Section 2.4 for the definition of the ideal function associated to an ideal).…”

Group-theoretic and topological invariants of completely integrally closed Prüfer domains

“…Following Vaughan and Yeagy [46], we say that a ring R for which every proper ideal is a product of radical ideals (i.e., "semi-prime" ideals) is an SP-ring. These rings have been studied by several authors; see for example [8,13,24,35,38,42,46,47] It is possible to construct a variety of SP-domains by using integral extensions of Dedekind domains. More precisely, one can obtain examples of SP-domains that are neither Dedekind domains nor Bézout domains by using the following method (which was already known to Grams [17]).…”

“…Following Vaughan and Yeagy [46], we say that a ring R for which every proper ideal is a product of radical ideals (i.e., "semi-prime" ideals) is an SP-ring. These rings have been studied by several authors; see for example [8,13,24,35,38,42,46,47]. An SP-domain is necessarily an almost Dedekind domain; that is, R M is a Dedekind domain for each maximal ideal M of R [46, Theorem 2.4].…”

“…As such they give existence in what might be considered the somewhat exotic setting of overrings of polynomial rings with as many variables as there are elements of the group. However, much of this same wide range of behavior can be seen in number-theoretic inspired contexts involving infinite towers of finite field extensions [8,17,22,24,31,32,43,46,47]. Interesting almost Dedekind domains also appear in the context of holomorphy rings [40].…”