Sorry in advance if this is too silly. Let $R$ be a right semihereditary ring and $P$ a projective right $R$-module. It is well-known that finitely generated (thus projective) submodules of $P$ form a lattice. Let now $F$ be a finitely generated submodule of $P$ and $S$ a (possibly infinitely generated) projective submodule of $P$. Is $F+S \subseteq P$ also necessarily projective? I'm mainly interested in the case of $R$ being noetherian, when $F \cap S$ is f.g. and thus projective (so in this case we know at least that $F+S$ has projective dimension $\leq 1$).
I would suspect that this is false, but I'm unable to create a counterexample. Thanks for any input.