Rings of Quotients of Rings of Functions
N. J. Fine, L. Gillman, J. Lambek

TL;DR
This paper explores the structure of rings of quotients of the ring of continuous functions on a space, showing how the maximal ring of quotients relates to functions on dense open sets, extending classical results.
Contribution
It applies Johnson and Utumi's rings of quotients to $C(X)$, characterizing the maximal ring of quotients as functions on dense open sets, generalizing classical quotient rings.
Findings
Maximal ring of quotients $Q(X)$ corresponds to functions on dense open sets.
In metric spaces, $Q(X)$ reduces to the classical ring of quotients.
Classical quotient ring is a proper subring of $Q(X)$ in general.
Abstract
From the original PREFACE: The rings of quotients recently introduced by Johnson and Utumi are applied to the ring of all continuous real-valued functions on a completely regular space . Let denote the maximal ring of quotients of ; then may be realized as the ring of all continuous functions on the dense open sets of (modulo an obvious equivalence relation). In special cases (e.g., for metric ), reduces to the classical ring of quotients of (formed with respect to the regular elements), but in general, the classical ring is only a proper sub-ring of .
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Topics in Algebra
