Exact sequence induces exact sequences for free parts and torsion parts?

Question

Let $A$ be a PID and consider the exact sequence of finitely generately modules over$A$:

$$0\longrightarrow M' \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}M''\longrightarrow 0 \tag{1}.$$

Denote the free part and torsion part by $F(M)$ etc. and $T(M)$ etc. respectively. Does the above exact sequence induces ones on the free parts and torsion parts?

Answer 1

The sequence $$ 0\rightarrow \mathbb Z\xrightarrow{n} \mathbb Z\rightarrow \mathbb Z_n\rightarrow 0 $$ is exact in $\mathbb Z\text{-}\mathsf{Mod}$. Passng to torsion we have $$ 0\rightarrow 0\rightarrow 0\rightarrow \mathbb Z_n\rightarrow 0 $$ which is not exact. Passing to free parts we have $$ 0\rightarrow\mathbb Z\xrightarrow{n}\mathbb Z\rightarrow 0\rightarrow 0 $$ which is also not exact.