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 ?
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.