Let $(R, m)$ be a Gorenstein local ring. $M$ is a finitely generated $R$-module with $\dim M=\dim R$ and $\text{pd }M<\infty$. Then $M$ contains a free module $R^k$, such that $$0\longrightarrow R^k\longrightarrow M\longrightarrow N\longrightarrow0$$ is exact with $\dim N<\dim M=\dim R$.
It seems to me that $R^k$ in this case should come from the torsion free part of $M$, but I have no idea how to show this. I also guess this result hold when $R$ is a Cohen-Macaulay ring.