The following is exercise 2.2.27 in Bruns and Herzog's book Cohen-Macaulay rings:
Let $R$ be a Noetherian ring such that every finite generated module has a finite free resolution. Then $R$ is UFD.
I don't know how to show that $R$ is integral domain. If this is true, using the proof of "regular local ring is UFD" in this book. Then the result follows.
Thank you in advance.