If $M$ is a finitely generated module and the quotient $M/T(M)$ by its torsion module $T(M)$ is projective, then $T(M)$ is also finitely generated.

Question

Let $M$ be an $R$-module and $T(M)$ be its torsion module. $M$ is finitely generated and $M/T(M)$ is projective. Without any further assumption on $R$, show that $T(M)$ is finitely generated.

The second condition seems to be a bit weird. I don't understand why it has to be projective. Does it mean we can use cohomological tools to solve it?