$M/im(f)$ is Torsion

Question

Let $R$ be a Principal ideal domain and $f:R^{n}\rightarrow M$, be an injective homomorphism where $M$ is a finitely generated $R$-module. I need to show that $M/im(f)$ is Torsion.

I know that since $f$ is injective then $ker(f)$ is trivial. Also that $im(f)$ is a submodule of $M$, therefore $M/im(f)$ is well defined since it is a quotient module. Do I have to use the existence of decomposition factors for $M$, and if so how?

Thanks

Answer 1

Let $M$ be $R\times R$ and let $f:R\to M$ be given by $f(r)=(r,0)$.

Then $M/Im(f)\cong R$ is not torsion.