Does anyone know of such a domain?



non-Dedekind Domain in which every ideal is generated by at most two elements
You may find Matlis' paper [The Two-Generator Problem for Ideals](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-17/issue-3/The-two-generator-problem-for-ideals/10.1307/mmj/1029000474.full) to be interesting, as its main theorem concerns the class of integral domains in which every ideal is generated by two elements.


It was proven by Cohen (in *Commutative Rings with Restricted Minimum Condition* ) that an integral domain with the property that there exists an integer $n$ such that every ideal can be generated with fewer than $n$ elements must be Noetherian and of Krull dimension 1.


Say that an integral domain R has property **FD** if every finitely generated torsion free R-module is direct sum of modules of rank 1. Moreover, say that R has property **FD locally** if RM has property FD for every maximal ideal M of R.


**Theorem** (simplified form) - Let R be an arbitrary integral domain. Then every ideal of R can be generated by two elements if and only if R is a noetherian ring that has property FD locally.


