  Blazar emission of gamma rays and cosmic ray production of gamma rays in
gas-rich clusters have been proposed recently as alternative sources of the
high energy extragalactic diffuse gamma ray background radiation. We show that
these sources also produce a very different high energy extragalactic diffuse
neutrino background radiation. This neutrino background may be detected by the
new generation of large neutrino telescopes under construction and may be used
to trace the origin of the extragalactic gamma radiation.
