Exact sequences and finitely generated projective modules

Question

Let $R$ be a ring and $A,B,C$ be $R$-modules in an exact sequence

$$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0.$$

Submodules or quotients of finitely generated projective modules are not necesserarily finitely generated and projective.

However, I was wondering if any assertion of the type "$A$ and $B$ are f.g. projective implies that $C$ is f.g. projective" is true (up to permutation of the names) ?

Answer 1

Counterexample: $R=A=B=\mathbb Z$, and the map from $A$ to $B$ is multiplication by 2. Then $C=\mathbb Z/2\mathbb Z$ is not projective.

Answer 2

On the positive side, if $B$ is finite projective and the sequence splits, then $A$ and $C$ are finite projective as well, because summands of finite projective modules are again finite projective. So, in particular, if $B$ and $C$ are finite projective, then $A$ is too (projectivity of $C$ ensures that the sequence splits).