.What confuses me is if g(N)=0, since an element x of M can be written as
No, it can't: there don't exist infinite sums here. N is the free module (i.e., the free abelian group) on , and every element there is a finite linear combination of the 's.
why then does it not follow that, since for each n, for any x in M? After that the purpose of the various k's is also a mystery to me.
Thanks for your help