Is there something non-trivial to be said about the subring of $K(X)$ spanned by one-dimensional bundles? 

If I am not mistaken, it is still a functor into commutative rings from the category of schemes. Moreover, it seems that there is some [interest][1] in the splitting problem for vector bundles, which goes back to Grothendieck's result that for $\mathbb{P}^1$ all bundles split. Also, by the splitting principle, any element of $K(X)$ lands in such a subring after a suitable pullback. 

But maybe my question is stupid for some obvious reason.


  [1]: https://www-users.cse.umn.edu/~mahrud/oral/exposition.pdf