Ideals in the ring of Colombeau generalized numbers

TL;DR

This paper explores the structure and properties of ideals in the ring of Colombeau generalized numbers, connecting them with various algebraic theories and analyzing their implications for nonstandard fields and topological modules.

Contribution

It provides new characterizations of different types of ideals and links the quotient rings to nonstandard fields, addressing open questions about prime ideals.

Findings

Characterization of prime, projective, pure, and topologically closed ideals.

Quotient rings modulo maximal ideals are isomorphic to nonstandard fields.

The Hahn-Banach extension property fails for many topological modules over these rings.

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.