Research Article

On quotient rings of pointfree function rings

Published in: Quaestiones Mathematicae
Volume 49 , issue 4 , 2026 , pages: 525–547

DOI: 10.2989/16073606.2025.2599364

Author(s): Batsile Tlharesakgosi University of South Africa, South Africa , Sharon Nyokabi Wamugunda University of South Africa, South Africa

Keywords: 06D22 , 13A15 , 54C30 , Completely regular frame , sublocale , quotient rings , classical rings of real-valued continuous functions , ideal , socle , essential maximal ideal

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Abstract

Let L be a completely regular frame. Associated with each I ∈ βL are two ideals, M^I and O^I , of the ring of real-valued continuous functions on a frame L. In this paper, we describe the intersection of all essential maximal ideals of an f-ring as the ideal M^A , with A equal to the join of all nowhere dense sublocales of βL. We show that every prime ideal of is essential if and only if A is a perfect sublocale. A ring is an EI N -ring if and only if L is an EF-frame. We show that is an EI N-ring if and only if the sublocale A (regarding A as a frame in itself ) is an EF-frame if and only if is an EI N-ring.

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