Despite reading about formal sums and especially the last comment in this post (which seems most relevant to my question) - I still feel the need to make sure I'm not missing something:
If there is a free abelian group generated by some basis $B := \{ b_1, b_2, \dots\}$ then all that is meant by "the elements in this group are formal sums" is just that:
- Adding any elements using the group operation "$+$" does not mean we get any other familiar element
- (my main issue) $\sum_i n_i b_i$ is just shorthand for applying the group operation, e.g. $2b_1-3b_2=2(b_1-b_2)-b_2$ since it's abelian, and so we can write it in terms of the inverse of $b_2$ this way (i.e. - we're just using the group operation)? Or is there something deeper here that I'm missing?