Characterizing integral domains for which every ideal, that can be generated by two elements, is projective?

Let $R$ be an integral domain. $R$ is called a Prufer domain if every finitely generated ideal of $R$ is projective. There are various equivalent conditions for $R$ being a prufer domain , in terms of ideal arithmetic (intersection distributes over sum ; product distributes over intersection etc. ), localization ($R_m$ is a valuation fomain for every maximal ideal $m$; $R_P$ is a valuation domain for every prime ideal $P$ ) , integral closure (every ring between $R$ and its fraction field is integrally closed ) , flatness etc. See the definition section here https://en.m.wikipedia.org/wiki/Prüfer_domain

My question is : Can we similarly give some characterization for those integral domain $R$ for which every ideal, that can be generated by two elements, is projective ? Has these type of domains been studied before ?

• I believe if every two generated ideal is projective, so is any finitely generated ideal, using the fact that finitely generated projective modules are locally free. Apr 1, 2018 at 15:16
• @Mohan : yes any projective module is locally free ... but how does that help here ? Could you please elaborate. Thanks
– user495643
Apr 1, 2018 at 20:08

First, one sees immediately that if $R$ has the property that all two generated ideals are projective, the same holds for any localizations.

Then one shows by induction as follows, any $n$-generated ideal is projective assuming the result for smaller $n$. Clearly we can assume $n>2$ and let $I=(x_1,\ldots, x_n)$ and then $J=(x_1,\ldots,x_{n-1})$ is projective by induction hypothesis and $I=J+Rx_n$. Since $J$ is projective, necessarily of rank one, we can localize and assume $J$ is free of rank one. But, locally, then $I$ is two generated and thus projective. So, $I$ is globally projective.

• Another argument : In $R_P$, any two generated ideal is projective, hence free, hence principal. So $R_P$ is a Bezout domain and also local, so $R_P$ is a valuation domain. Thus $R$ is locally a valuation domain which is equivalent to saying $R$ is Prufer.
– user495643
Apr 2, 2018 at 8:48