  We define a faithful contravariant functor NCSpec from the category of rings
to the category of ringed spaces, and show that if R is a commutative ring then
NCSpec(R) may be viewed as a completion of Spec(R) in an appropriate sense. We
then explain how the spaces NCSpec(R) may be glued, and study quasicoherent
sheaves on them. As an example, we compute the category of quasicoherent
sheaves on a space constructed from a skew-polynomial ring R by an analogue of
the Proj construction.
