[Paper Review] Rings of Quotients of Rings of Functions

N. J. Fine, Leonard Gillman+1 more|arXiv (Cornell University)|Mar 6, 2024

Rings, Modules, and Algebras92 citations

TL;DR

The paper studies maximal rings of quotients for the ring C(X) of real-valued continuous functions on a completely regular space X, showing Q(X) can be realized as continuous functions on dense open sets modulo an equivalence, and illustrating that in general the classical ring of quotients is a proper subring of Q(X).

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 $Q(X)$.

Motivation & Objective

• Motivate the study of maximal rings of quotients in relation to C(X).
• Generalize the construction of quotients from metric spaces to completely regular spaces.
• Describe how Q(X) can be realized as functions on dense open subsets of X.
• Compare Q(X) with the classical ring of quotients in general versus special cases.
• Highlight the relationship between topology of X and algebra of C(X).

Proposed method

• Apply the rings of quotients recently introduced by Johnson and Utumi to C(X).
• Define Q(X) as the maximal ring of quotients of C(X).
• Realize Q(X) as the ring of all continuous functions on dense open sets of X, modulo a natural equivalence relation.
• Show that in special cases (such as metric X) Q(X) coincides with the classical ring of quotients formed with respect to regular elements.
• Demonstrate that the classical quotient ring embeds as a proper subring of Q(X) in general.

Experimental results

Research questions

• RQ1 What is the maximal ring of quotients of C(X) for a completely regular space X?
• RQ2 How can Q(X) be realized concretely in terms of continuous functions on dense open subsets?
• RQ3 When does Q(X) coincide with the classical ring of quotients, and what happens in general?
• RQ4 What is the relationship between metric properties of X and the structure of the quotient rings?
• RQ5 How does the classical ring of quotients relate to the maximal ring of quotients in this setting?

Key findings

• Q(X) can be realized as the ring of all continuous functions on dense open sets of X modulo an equivalence relation.
• In metric spaces X, Q(X) reduces to the classical ring of quotients formed with respect to regular elements.
• In general, the classical ring of quotients is only a proper subring of Q(X).
• The maximal ring of quotients extends beyond the classical construction in a topological context.
• The study applies Johnson and Utumi’s quotient framework to rings of real-valued continuous functions.
• The paper provides a long, self-contained exploration of these rings in the setting of completely regular spaces.