Research Article
On quotient rings of pointfree function rings
DOI:
10.2989/16073606.2025.2599364
Keywords:
06D22, 13A15, 54C30, Completely regular frame, sublocale, quotient rings, classical rings of real-valued continuous functions, ideal, socle, essential maximal ideal,
Abstract
Let L be a completely regular frame. Associated with each I ∈ βL are two ideals, MI
and OI
, 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 MA
, 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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