Is it true that if a module has a free resolution of length $d$ then any of its submodule has a free resolution of length $\leq d$?



Projective dimension
No. A submodule of a free module need not have finite projective dimension.
As a simple example let $R=\mathbb{Z}/p^2\mathbb{Z}$. The free module $R$
has a submodule $p\mathbb{Z}/p^2\mathbb{Z}\cong\mathbb{Z}/p\mathbb{Z}$
which has no finite projective resolution.


