What does mean by the sentence "an $R$-module $M$ is $x$-torsion free", for $x \in R$ a fixed element?
I got some information as follows:
If $M$ is $x$-torsion free if for each $m,n \in \mathbb{N}$, we have a short exact sequence like $$ 0 \to M/x^n M \xrightarrow{x^m} M/x^{m+n}M \xrightarrow{?} M/x^mM \to 0.$$
Is it true?
If this is true, what is 2nd map (?) above?
How to prove it?