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 \cite {Coykendall}. In \cite {Grams}, 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.