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.

Edit: Some concrete questions:

 1. Is the existence of an indecomposable bundle on $X$ imply that the subring generated by line bundles is proper?
 2. Is there an analog of higher $K$-groups for this definition? (I don't see any problem with taking the usual motivic construction, but what do I know)
 3. Are there known computations of this ring?
 4. Is there a general relationship between the usual $K$-ring and this subring? 

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