I am trying to understand two statements in Serre's Local Fields:
Let V be a finite dimensional vector space over K. A lattice of V (with respect to A) is a sub-A-module X of V that is finitely generated and spans $V$
The first:
If A is principal, this means that $X$ is a free A-module of rank $[V: K]$;
Where does the rank $[V: K]$ come from? and why is X reduced to its torsion free part?
The second:
one can often reduce to this case by localisation, i.e., by replacing A with $A_p$ and $X$ with $A_pX$
I don't see how?
Thanks for your help.