TITLE: Example of FD not satisfying ACCP
QUESTION [10 upvotes]: Let $A$ be an integral domain. We know that if $A$ satisfies ACC on principal ideals then it is a Factorisation Domain. I want to know about the converse, i.e., 

Does there exist a FD not satisfying ACC on principal ideals?

Edit. A Factorisation Domain is an integral domain whose every non-zero non-unit element is a finite product of irreducible elements.
Thank You.

REPLY [3 votes]: Here is another construction from Roitman's paper "Polynomial Extensions of Atomic Domains", which has some additional interesting properties.
For a domain $D$ and $S \subseteq D\setminus(0)$, define $L(D;S) := D[\{X_s,sX_s^{-1}\}_{s \in S}]$, where $\{X_s\}_{s \in S}$ is a family of algebraically independent indeterminates.   (Note the similarity here to a Laurent polynomial ring.)  Now, starting from any domain $A_0$ that is not a weak GCD domain, recursively define $A_n := L(A_{n-1}; S_{n-1})$ for $n \ge 1$, where $S_{n-1}$ is the set of reducible elements of $A_{n-1}$.  Define $A_\infty:= \bigcup_{n=0}^\infty A_n$.
Roitman shows that $A_\infty$ is atomic but $A_\infty[X]$ is not.  Since it is well-known that the ACCP property is preserved by polynomial extensions, it follows that $A_\infty$ is atomic but fails the ACCP.  Another interesting property of $A_\infty$ related to the "elasticity" of factorizations:  every reducible element has arbitrarily long atomic factorizations, but also has an atomic factorization of length 2.  Variants of Roitman's construction have since been used to provide a lot of other interesting factorization examples.
Edit:  I added in a part in bold that I originally left out.  A domain is a weak GCD domain if every pair of nonzero elements $a,b$ has a maximal common divisor c, meaning that $(c)$ is a minimal element of the set of principal ideals containing $(a,b)$.  For example, the domain $\mathbb Q[\{Z,X/Z^n,Y/Z^n\}_{n=0}^\infty]$ is not a weak GCD domain, since $X$ and $Y$ have no maximal common divisor.  The domain $A_\infty$ is always atomic regardless of which domain you start with; the other properties stated require some sort of assumption about $A_0$.