$(1)$ Why do we assume a module $M$ over a commutative ring $R$ to be $p$-torsion free, in some cases ?
How does it help us or how does it cost us if we don't assume it?
We know that if $M$ is $p$-torsion free module, then $pm=0 \Rightarrow m=0, \ \forall m \in M$.
So what is the benefit of assuming the above condition ?
We don't assume modules are $p$-torsion-free unless (1) we're proving something that's false in the presence of $p$-torsion, or (2) we're proving something that's considerably more difficult to prove in the presence of $p$-torsion, or (3) we're proving something that's trivial in the presence of $p$-torsion, or (4) we're not interested in the case where there's $p$-torsion.