# On a special type of Noetherian regular rings

Let $$R$$ be a commutative Noetherian ring having the property that for every $$R$$-module $$M$$ that has finite projective dimension, every submodule of $$M$$ also has finite projective dimension. Then $$R$$ is obviously regular since every ideal has finite projective dimension.

My question is: Does $$R$$ have finite global dimension ?

If $$R$$ has infinite global dimension, then it has a module $$X$$ of infinite projective dimension. But if $$P_0\to X$$ is a surjective map from a projective module to $$X$$, with kernel $$Y$$, then $$Y$$ must have infinite projective dimension, since if $$Y$$ had a finite projective resoltion $$\cdots\to 0\to P_n\to\cdots\to P_2\to P_1\to Y\to0$$ then $$X$$ would have a finite projective resolution $$\cdots\to 0\to P_n\to\cdots \to P_1\to P_0\to X\to0.$$
So $$Y$$ is a module of infinite projective dimension that is a submodule of $$P_0$$, which is projective and so certainly has finite projective dimension.