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.

We construct a norm on the nonzero elements of a Prüfer domain and extend this concept to the set of ideals of a Prüfer domain. These norms are used to study factorization properties Prüfer of domains.

No abstract

Given a ring $R$, an ideal $I$ of $R$,  and an element $a\in I$,  we say $a=\lambda b_1\cdots b_k$ is a $\tau_I$-factorization of $a$ if $\lambda$ is any unit and $b_1\equiv\cdots\equiv b_k\pmod{I}$.  In this paper, we investigate the $\tau_I$-atomicity of PIDs with ideals where $R/I$ has size four.

Characterizations of bounded and finite factorization domains are given using topological notions. Using our characterizations, the almost Dedekind domain and Prüfer domain constructed by Grams [3] are shown to be a BFD and an FFD, respectively. For a class of almost Dedekind (not Dedekind) domains it is shown that satisfying the ascending chain condition for principal ideals implies BFD.