# Noetherian almost Dedekind domain

A Dedekind domain is an integral domain in which every nonzero proper ideal factors into a product of prime ideals, and an integral domain $R$ is called almost Dedekind whenever $R_m$ is Dedekind domain for every maximal ideal $m$ of $R$. It is well known that a Dedekind domain is noetherian and so it's maximal spectrum is noetherian space as a subspace of Zariski topology. Is there any reference for the following statment?

An almost Dedekind domain with noetherian maximal spectrum is a Dedekind domain.

• In "Almost Dedekind domains which are not Dedekind", Multiplicative ideal theory in commutative algebra, 279–292, Springer, New York, 2006, K. Alan Loper says: "A domain $D$ is a Prufer domain if $D_M$ is a valuation domain for each maximal ideal $M$ of $D$. The Noetherian Prufer domains are the Dedekind domains. It follows that if $D$ is a Dedekind domain, then $D_M$ is a Noetherian valuation domain for each maximal ideal $M$ of $D$. A domain $D$ is almost Dedekind if $D_M$ is a Noetherian valuation domain for each maximal ideal $M$ of $D$. (cont)" Apr 18 '16 at 18:41
• (cont) "Clearly, Dedekind domains are almost Dedekind. The point of the designation is that almost Dedekind domains satisfy the characterization given above of Dedekind domains, except they are not assumed to be Noetherian." This would suggest that not only is the statement true, but in fact it was the impetus behind the definition. Apr 18 '16 at 18:43
• (Above comments refer to the question as it was before being edited) Apr 18 '16 at 19:17

In any ring, the ACC on radical ideals is the same as satisfying (i) the ACC on prime ideals and (ii) every element has only finitely many minimal primes. Since almost Dedekind domains are one-dimensional, we therefore have J-Noetherian (i.e., Noetherian maximal spectrum) $\Leftrightarrow$ ACC on radical ideals $\Leftrightarrow$ every nonzero element is only contained in finitely many maximal ideals. With this in mind, we can proceed to show that every nonzero ideal $I$ of a J-Noetherian almost Dedekind domain $D$ is 2-generated, so $D$ is Dedekind. Pick $0 \ne x \in I$. Since $D/(x)$ has only finitely many maximal ideals and $I/(x)$ is a locally principal ideal of $D/(x)$, it follows that $I/(x)$ is principal (for details, see Lemma 1 in "Multiplication ideals, multiplication rings, and the ring R(X)" by D.D. Anderson). Therefore $I$ is 2-generated.