Converse of fundamental theorem of finitely-generated modules

Question

Let $R$ be a ring. If every finitely-generated $R$-module $M$ is isomorphic to a finite direct product of quotients of $R$ by ideals then call $R$ a wheel ring. For a domain $R$ we have the implications

$$\rm PID\implies wheel\implies Bezout $$

The $1$st implication is the content of the fundamental theorem of f.g. modules over a PID. The second implication is equivalent to its contrapositive, $\neg\rm Bezout\implies\neg wheel$: suppose $R$ is a wheel domain but not Bezout. Let $I$ be a f.g. non-PI. Since the ideal $I$ is torsionfree and $R$ is wheel, $I$ must be free over $R$, and since $I$ is not principal it must have rank $>1$, so there must exist $a,b\in R$ such that $Ra+Rb$ is direct. But $(-b)a+(a)b=0$, so it can't be direct, which is a contradiction.

Consider the reversal $\rm Bezout\Rightarrow wheel\Rightarrow PID$. Bezout is a strictly weaker property than PID, so at least one of the reverse implications must be false. Which one, or both, fails to hold? If we have $\rm wheel\not\Rightarrow PID$, then what is an example of a wheel domain which is not a PID? If on the other hand $\rm Bezout\not\Rightarrow wheel$, then what is an example of a Bezout domain which is not wheel?

Answer 1

The "wheel" rings described in my question are called FGC rings in the literature. There is a complete classification of the commutative FGC rings, and some work on noncommutative theory. The characterization is that a commutative ring is FGC iff it is a direct sum of maximal valuation rings, almost maximal Bezout domains, and "torch" rings. I don't presently know what these are, but apparently they're described in Commutative Rings Whose Finitely Generated Modules Decompose (Brandal).

Another similar notion related to me in chat by Ted Shifrin (one of his colleagues did some kind of work on the topic) is that of an elementary divisor domain. The usual proof of the fundamental theorem of finitely-generated modules over PIDs uses Smith Normal Form, and a commutative ring is an EDD precisely if matrices still admit SNFs. This is equivalent to all of its f.g. modules admitting an elementary divisor decomposition, which is stronger than being a direct sum of cyclic submodules, as in an FGC ring we could still have factors that look like $R/I$ for some infinitely-generated nonprincipal ideal $I$ (which is not an elementary divisor, where $I$ is principal).