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 ?
For any ring $R$, if there is an $R$-module $N$ of infinite projective dimension then there is a projective $R$-module (which certainly has finite projective dimension!) with a submodule of infinite projective dimension.
Indeed, choose an epimorphism from a projective module $P$ to $N$, and let $K$ be the kernel, so we have a short exact sequence $$0\longrightarrow K\longrightarrow P\longrightarrow N\longrightarrow0.$$
Since $N$ has infinite projective dimension and $P$ is projective, $K$ also has infinite projective dimension.