We know that for a PID $R$, any finitely generated module is of the form $\frac{R}{(a\_1)} \oplus \dots \oplus \frac{R}{(a\_s)} $.
I was wondering if the converse of this statement is true, that is, is it true that for a domain $R$, if any f.g. module is isomorphic to $\frac{R}{I\_1} \oplus \dots \oplus \frac{R}{I\_s}$ for ideals $I\_i \triangleleft R $, must $R$ be a PID? 


I know that a counter example to the above statement must be non Noetherian (and infact, it must be that any non-principal ideal is not f.g.) because if $I = (a\_1, ..., a\_n)$ is a non principal f.g. ideal, that it is a torsion free $R$ module, but it isn't free, since $x\_1, ... x\_m$ are not linearly independent if $m > 1$ since $x\_2 x\_1 + (-x\_1)x\_2 = 0$, hence they do not form a basis. 


PS: I would generally like $R$ to be commutative unitial and an integral domain, but I would love to know even for other cases. 



Inverse of the Structure Theorem for Finitely Generated Modules over PID
There seems to be quite some literature about rings with this property, sometimes under the name "FGC domains".


From Googling, not personal knowledge:


In Theorem 14 of
*Kaplansky, Irving*, [**Modules over Dedekind rings and valuation rings**](http://dx.doi.org/10.2307/1990759), Trans. Am. Math. Soc. 72, 327-340 (1952). [ZBL0046.25701](https://zbmath.org/?q=an:0046.25701), it is proved that an almost maximal valuation domain has this property.


"Maximal" means that a system $x\equiv a\_i\pmod{I\_i}$ of congruences has a solution if each pair of the congruences has a solution, and "almost maximal" is the same, but only for such systems of congruences with $\bigcap\_iI\_i\neq0$. I'm not sure what the simplest example is that is not a PID, but I guess the completion of $R=k[x^\alpha\mid 0<\alpha\in\mathbb{Q}]$ with respect to the set of ideals $x^\alpha R$ is a maximal valuation domain.


Osofsky constructed an example of an FGC domain that is neither a PID nor a valuation domain, and work of many people, culminating in
*Brandal, Willy; Wiegand, Roger*, [**Reduced rings whose finitely generated modules decompose**](http://dx.doi.org/10.1080/00927877808822241), Commun. Algebra 6, 195-201 (1978). [ZBL0368.13005](https://zbmath.org/?q=an:0368.13005), led to the theorem that FGC domains are precisely the almost maximal Bezout domains.


And there's a whole Springer Lecture Notes volume by Brandal on the not-necessarily-domain case:


*Brandal, Willy*, Commutative rings whose finitely generated modules decompose, Lecture Notes in Mathematics. 723. Berlin-Heidelberg-New York: Springer-Verlag. 116 p. (1979). [ZBL0426.13004](https://zbmath.org/?q=an:0426.13004).


