Ideals in the Ring of Colombeau Generalized Numbers

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 [1]. Also z-ideals in the sense of [21] 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.

Recent progress in special Colombeau algebras: geometry, topology, and algebra

“…For details and proofs we refer to the original sources [1,2,29]. In what follows, topological properties always refer to the sharp topology onK (cf.…”

“…We conclude this section with the following interesting connection to the nonstandard space of asymptotic numbers (cf. [24]), established in [29], Th. 7.2:…”

“…In particular, a thorough study of algebraic properties was carried out (e.g., [1,29]) and topological and functional analytic structures on Colombeau spaces were developed and refined to a high degree (e.g., [25,26,9,10]). …”

Asymptotic Ideals (Ideals in the Ring of Colombeau Generalized Constants with Continuous Parametrization)

“…We denote I ⊳ K cnt for a proper ideal I of K cnt (i.e., I = K cnt ). K is an exchange ring [13], i.e., for each a ∈ K, there exists an idempotent e ∈ K such that a + e is invertible. Unlike K, K cnt is not an exchange ring [5,Lemma 4.3].…”

“…In this way, we obtain a classification of maximal and minimal prime ideals in terms of maximal and prime filters. The methods used in this paper are inspired by the study of the ideals in K [1,13] and by the study of maximal ideals of rings of continuous functions by Gillman and Jerison [7]. Compared to [7], the main novelty is the adaptation to the asymptotic nature of the ring K cnt .…”