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 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.