  We shall describe a simple generalization of commutative rings. The category
GR of such "rings", contains the ordinary commutative rings (fully faithfully),
but also the "integers" and "residue field" at a real or complex place of a
field ; the "field with one element" (the initial object of GR ); the
"arithmetical surface" ( the sum in the category GR of the integers with them
self: Z(x)Z ) . We shall show that this geometry "see" the real and complex
places of a number field (there is an Ostrowski theorem that the valuation
sub-GR of a number field correspond to the finite and infinite primes; there is
a compactification of the spectrum of the integers ). One can develop algebraic
geometry using GR following Grothendieck paradigm . Quillen's homotopical
algebra replaces homological algebra . There is an analogue of Quillen's
cotangent bundle - in particular there is a theory of non-additive derivations,
with modules of Kahler differentials which satisfy all the usual exact
sequences - we compute explicitly the differentials of the integers Z over the
initial object F1 . Finally we associate with any topological compact valuation
GR a meromorphic function - its "zeta" : for the p-adic integers we get the
p-local factor of zeta, for the "real integers" we get the gamma factor.
