  In this paper, we introduce a new algebraic type of `convexoid rings', and we
give the definition of (weak) convexoid schemes, which share similar properties
with ordinary schemes. As a result, we give a purely-algebraic construction of
the compactification of Spec Z (in Arakelov's sense) which is realized as the
Zariski-Riemann space in the category of weak convexoid schemes.
