Why are you involving the rationals? If ax + by is zero for some non-zero rationals a and b, there also are non-zero integers i and j such that ix + jy is zero (multiply a and b by the product of their denumerators to get one such pair)
Number systems with a well behaved division operation (fields) are heavily studied and have lots of wonderful properties. If we allow for division, thus turning the integers into the rationals (which form a field), then we have access to all this.
So, while you are correct, to a mathematician it might be simpler to start with a field and then be able to wave our hands at centuries of work in field theory and just say, "All that stuff applies here."
In particular, the notion of linear independence is generally defined only for subsets of a vector space. And being a vector space requires that the scalars (allowable coefficients) form a field.
