Let $R$ be a regular ring i.e. a commutative Noetherian ring whose localisations at every prime ideal is regular local ring. Then every finitely generated $R$-module has finite projective dimension, however not every $R$-module may have finite projective dimension.
My question is: Let $M$ be an $R$-module of finite projective dimension. Then, does every submodule also have finite projective dimension ? If this is not true in general, what if we also assume $R$ is an integral domain ?