Conditions on a Module Implied by $\mathrm{Ext}^1 = 0$

Question

Let $R$ be a Noetherian ring, and let $M$ be a finitely generated $R$-module. If $M$ is projective, and in particular if $M$ is free, then it is not hard to see that $\mathrm{Ext}^1(M,R) = 0$. Is there some kind of converse to this statement? In other words, what can we say about the module $M$ given that $\mathrm{Ext}^1(M,R) = 0$?

Answer 1

Usually the converse is given as follows. An $R$-module $M$ is projective if and only if $\mathrm{Ext}^1_R(M,N)=0$ for all $R$-modules $N$. If we only have $\mathrm{Ext}^1_R(M,R)=0$, this need not be true. Sometimes it is undecidable, for example consider $R=\Bbb{Z}$ and a $\Bbb{Z}$-module $M$ with $\mathrm{Ext}^1_{\Bbb{Z}}(M,\Bbb{Z})=0$ (Whitehead problem).