Rings of Quotients of Rings of Functions

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 $C(X)$ of all continuous real-valued functions on a completely regular space $X$. Let $Q(X)$ denote the maximal ring of quotients of $C(X)$; then $Q(X)$ may be realized as the ring of all continuous functions on the dense open sets of $X$ (modulo an obvious equivalence relation). In special cases (e.g., for metric $X$), $Q(X)$ reduces to the classical ring of quotients of $C(X)$ (formed with respect to the regular elements), but in general, the classical ring is only a proper sub-ring of $Q(X)$.