IDEALS IN THE RING OF COLOMBEAU GENERALIZED NUMBERS

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 [1]. Also z-ideals [23] are characterized. It is shown that the quotient rings modulo maximal ideals are canonically isomorphic with nonstandard fields of asymptotic numbers and that the Hahn-Banach extension property does not hold for a large class of topological modules over the ring of Colombeau generalized numbers.