Let $M$ be a non-zero, finitely generated projective module over a commutative integral domain $R$.
Is it necessarily true that there is an $R$-linear surjection $M\to R$ ? If this is not true in general, what if we also assume $R$ is Noetherian ?
I know this is true if $R$ is local as projective modules over local rings are free.
Of-course this is not true if we drop the integral domain assumption on $R$, for example $M=\mathbb Z/2\times 0$ is a finitely generated projective module over the Noetherian ring $R=\mathbb Z/2\times\mathbb Z/2$, but definitely $M$ cannot surject onto $R$ for cardinality reason. But with $R$ an integral domain, I have no counterexample.