If every R-module is projective, then R is a field.

If every R-module is projective, then R is a field.

By the way, we're assuming R is an integral domain. I'm guessing we're
going to want to show that R has no nontrivial proper ideals. So, let I be
an ideal in R.
$0\rightarrow I \rightarrow R\rightarrow R/I\rightarrow 0$ splits, since
R/I is an R-module, thus projective. so R is isomorphic to RxR/I, but I'm
not too sure what to do from there.
Alternatively, every R-module is projective iff every R-module is
injective, so Baer's criterion might be useful, but again I'm not sure
where to go from that.
Any hints?