Let's say we have $f:A \to B$ where $A$ and $B$ are R-modules and $f$ is a R-module homomorphism.
If A has a torsion, then $\{a \in A \mid ar=0 \text{ for some nonzero } r \in R\}\neq \varnothing $. Let's say $a \in A$ and $0 \neq r \in R$ are such that $ar=0$. Then $rf(a)=f(ra)=f(0)=0$. so $f(A)$ has a torsion.
Therefore, can we conclude that
"If $f(A)$ is torsion free, then $A$ is torsion free."?