  A Laurent polynomial ring $A[t,1/t]$ with coefficients in a unital ring $A$
determines a category of quasi-coherent sheaves on the projective line over
$A$; its $K$-theory is known to split into a direct sum of two copies of the
$K$-theory of $A$. In this paper, the result is generalised to the case of an
arbitrary strongly $\mathbb{Z}$-graded ring $R$ in place of the Laurent
polynomial ring. The projective line associated with $R$ is indirectly defined
by specifying the corresponding category of quasi-coherent sheaves. Notions
from algebraic geometry like sheaf cohomology and twisting sheaves are
transferred to the new setting, and the $K$-theoretical splitting is
established.
