Could someone please give me a hint for this problem?
The following sequence is exact:
$0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \xrightarrow{h} D \xrightarrow{i} E \xrightarrow{j} F \to 0$.
Prove: If $A, B, D, E$, and $F$ are free R-modules, then $C$ is torsion-free. (R is a PID, particularly: $\mathbb{Z}[1/p]$.)
This is my attempt:
$0 \rightarrow \text{coker}f \rightarrow C \rightarrow \text{im}h \to 0$.
I know $\text{im}h$ is torsion-free since it is a subgroup of a free group. I know that cokernel of $f$ is B/im$f$. And if k$\sigma$ is an element of im$f$, then showing that $\sigma$ is an element of im$f$ would prove the result, where k is an element of the ring.