TITLE: A question about Cohen-Kaplansky domains
QUESTION [6 upvotes]: Let $A$ be a domain. Recall that $A$ is Cohen-Kaplansky (or CK) if 
(CK) any nonzero nonunit of $A$ is a product of irreducible elements, and there are only finitely irreducible elements up to multiplication by units.
Consider the following condition on $A$:
(RFFF) the field of fractions $K$ of $A$ is ring-finite over $A$, that is, $K$ is a finitely generated $A$-algebra.
[RFFF stands for "ring-finite field of fractions".]
Clearly, (CK) implies (RFFF).
Naive Conjecture: (RFFF) implies (CK).
Question: Is the Naive Conjecture true?
One easily sees that (RFFF) is equivalent to 
(RFFF') there is a nonzero element $a$ in $A$ such that, for any $b$ in $A$, there is a positive integer $n(b)$ such that $b$ divides $a^{n(b)}$.
In particular, the Naive Conjecture holds for unique factorization domains, and also for Dedekind domains.
EDIT A. Thank you very much to Hagen and Bill Dubuque for their answers! Here is an update. From now on, following Kaplansky, I'll call G-domain what I called RFFF-domain above. 
$(1)$ Here is how I understand a part of Hagen's answer: Bourbaki, Algèbre Commutative, VI.$6.3$, provides the following example of a G-domain which is not a CK-domain. There exist a field $K$ and a surjective multiplicative monoid morphism $x\mapsto|x|$ from $K$ onto $\mathbb Q_{\ge0}$ satisfying
$$
|x|=0\iff x=0,\quad|x+y|\le\max(|x|,|y|),
$$
such that the closed ball of radius one is a local domain $A$ with maximal ideal $\mathfrak m$ equal to the open ball of radius one, the ideals of $A$ being the closed balls of radius $r$ with $0\le r\le1$ and the open balls of radius $r$ with $0 < r\le1$, and the group $A^\times$ of units of $A$ being the sphere of radius one. In particular $0$ and $\mathfrak m$ are the only prime ideals. Thus any nonzero $a$ in $\mathfrak m$ satisfies $A[a^{-1}]=K$, so $A$ is a G-domain. The group $K^\times/A^\times$, being isomorphic to $\mathbb Q_{ > 0}$, is not finitely generated, that is $A$ is not a CK-domain.
$(2)$ In the article 
Rings with a finite number of primes, I. S. Cohen and Irving Kaplansky, Trans. Amer. Math. Soc. $60$ ($1946$), $468$-$477$,
it is proved that CK-domains are noetherian (Theorem $6$ p. $471$). 
$(3)$ Recall from Bill Dubuque's answer that a domain $A$ is a noetherian G-domains if and only if $A$ has only a finite number of non-zero prime ideals, all of which are maximal. 
$(4)$ The main question which remains open (at least for me) is this: 
Is there a noetherian G-domain which is not a CK-domain?
That is:
Is there a noetherian domain $A$ such that 
$\bullet\ A$ has only a finite number of non-zero prime ideals, all of which are maximal,
$\bullet\ A$ has infinitely many (association classes of) irreducibles?

REPLY [6 votes]: A less involved example of a Noetherian G-domain that is not a CK-domain: 
Let $\mathbb R$ and $\mathbb C$ denote the fields of real and complex numbers respectively and consider $S = R + X\mathbb C[[X]]$ the ring of power series over $\mathbb C$ with real constant terms. Since the only nonunits of $S$ are powers of $X$ (with coefficients from $\mathbb C$) it is easy to see that 
(1) $X\mathbb C[[X]]$ is the maximal ideal of $S$, 
(2) $S$ is a G-domain and is $1$-dimensional. 
Next note that $X\mathbb C[[X]] = XS+iXS$, where $i$ is the imaginary number. So $S$ is Noetherian. Now a CK-domain is an atomic domain that contains only a finite number of irreducible elements up to associates. But $S$ has infinitely many non-associated atoms: $(r + i)X$ where $r$ ranges over real numbers. (This is a simplified version of what follows Corollary 3.6 of [AM]. )
It has already been established, in a simple enough fashion, here that there are non-Noetherian G-domains and so (RFF) does not imply CK. But an example attributed to Bill Dubuque sort of attracted my attention. The example is: If a valuation domain $V$ has a nonzero minimal prime ideal $P$ then $V[1/x] =qf(V)$ for all nonzero $x\in P$. I have the following observation. (Incidentally, where is Bill Dubuque? Is he Gone?) 
Observation: Let $D$ be a domain with a minimal nonzero prime ideal $P$ which is comparable with each principal ideal of $D$, then $D$ is a G-domain.
While it gives Bill’s example it delivers examples that may not be easy to see. 
Example B: The ring $T= \mathbb Z+X\mathbb Q[[X]]$ is a G-domain with infinitely many maximal ideals. 
Example C: The ring $\mathbb Z+X\mathbb R[[X]]$ is a G-domain with infinitely many maximal ideals.
Next the statement in (3) of Edit A seems a little I have a feeling that “a domain $A$ is a noetherian G-domain if and only if $A$ has only a finite number of non-zero prime ideals, all of which are maximal” needs to be changed to “a noetherian domain $A$ is a G-domain if and only if $A$ has only a finite number of non-zero prime ideals, all of which are maximal”.
Now a bit about the history tidbits by Gone (er Bill, how the heck am I to know?). If you really look up [AM] you would know that the logical next step was [AAZ], the study of atomic domains with almost all atoms prime. Unfortunately there was a massive mistake introduced in the paper, in an effort to get more bang without putting in any bucks. The mistake was caught by Martine Picavet-L’Hermite in [MP]. After a massive ignoring campaign there appeared [CA] where, the mistake was owned. My sympathies go to Martine, she is one of the folks who come up with something remarkable and have to face blank faces of the confirmed geniuses. 
[AAZ] D.D. Anderson, D.F. Anderson and M. Zafrullah, (1992). Atomic domains in which
almost all atoms are prime. Comm. Algebra 20:1447–1462.
[AM] D.D. Anderson, J. Mott, (1992). Cohen–Kaplansky domains: Integral domains
with a finite number of irreducible elements. J. Algebra 148:17–41.
[CA] S. Chun and D.D. Anderson, A class of atomic rings, Comm. Algebra 40(2012) 1086-1095. 
[MP] Picavet-L’Hermitte, M. (2000). Factorization in Some Orders with a PID as Integral
Closure, Algebraic Number Theory and Diophantine Analysis. Berlin: de Gruyter,
pp. 365–390.