Structure spaces and allied problems on a class of rings of measurable functions

2024/08/01 by Dey, Soumajit, Acharyya, Sudip Kumar, Mandal, Dhananjoy
#46E30 #54C40 #FOS: Mathematics #General Topology (math.GN) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2408.00505

Abstract

A ring S(X,A) of real valued A-measurable functions defined over a measurable space (X,A) is called a χ-ring if for each E∈ A , the characteristic function χ_E∈ S(X,A). The set U_X of all A-ultrafilters on X with the Stone topology τ is seen to be homeomorphic to an appropriate quotient space of the set M_X of all maximal ideals in S(X,A) equipped with the hull-kernel topology τ_S. It is realized that (U_X,τ) is homeomorphic to (M_S,τ_S) if and only if S(X,A) is a Gelfand ring. It is further observed that S(X,A) is a Von-Neumann regular ring if and only if each ideal in this ring is a Z_S-ideal and S(X,A) is Gelfand when and only when every maximal ideal in it is a Z_S-ideal. A pair of topologies u_μ-topology and m_μ-topology, are introduced on the set S(X,A) and a few properties are studied.

Related

• Ideals in Rings and Intermediate Rings of Measurable Functions
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• Maximal ideals in rings of real measurable functions
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• Recent progress in Rings and Subrings of Real Valued Measurable\n Functions