  For certain subrings of the mod-p cohomology ring of a compact Lie group, we
give a description of the prime ideal spectrum, analogous to Quillen's
description of the spectrum of the whole ring. Examples of such subrings
include the Chern subring (the subring generated by Chern classes of all
unitary representations), and for finite groups the subring generated by Chern
classes of representations realizable over any specified field.
  As a corollary, we deduce that the inclusion of the Chern subring in the
cohomology ring is an F-isomorphism for a compact Lie group G if and only if
the following condition holds: For any homomorphism f between elementary
abelian p-subgroups of G such that f(v) is always conjugate to v, there is an
element g of G such that f is equal to conjugation by g.
