  In this paper, the structure of the ideals in the ring of Colombeau
generalized numbers is investigated. Connections with the theories of exchange
rings, Gelfand rings and lattice-ordered rings are given. Characterizations for
prime, projective, pure and topologically closed ideals are given, answering in
particular the questions about prime ideals in [Aragona-Juriaans]. Also
z-ideals in the sense of [Mason] are characterized. The quotient rings modulo
maximal ideals are shown to be canonically isomorphic with nonstandard fields
of asymptotic numbers. Finally, a detailed study of the ideals allows us to
prove that (under some set-theoretic assumption) the Hahn-Banach extension
property does not hold for a large class of topological modules over the ring
of Colombeau generalized numbers.

Rings and Algebras