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arXiv · 0707.0698

Ideals in the ring of Colombeau generalized numbers

Hans Vernaeve
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Abstract

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.

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Hans Vernaeve. 2007-07-04. Ideals in the ring of Colombeau generalized numbers. https://arxiv.org/abs/0707.0698

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