# Rings of Quotients of Rings of Functions

## N.J. Fine, L. Gillman, and J. Lambek

Michael Barr and Robert Raphael have prepared a version of the original 1965 book [McGill University Press, 1966, v + 97 pp.]. Here is a PDF file (72 pages, 495 Kb).

The original book was reviewed by W.W. Comfort for Zentrallblatt für Mathematik und ihre Grenzgebiete. Here are the scanned images (Zbl 0143.35704) provided by Zentralblatt MATH.

Here are the Preface by Fine, Gillman and Lambek and the Editors' Preface by Barr and Raphael.

### Preface

These seminar notes contain the results announced in Abstracts 576-219, Notices Amer. Math. Soc. 7 (1960), 980; 61T-27, ibid. 8 (1961), 60; and 61T-28, ibid. 8 (1961), 61.

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 in 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 subring of Q(X).

A natural metric may be imposed on the ring Q(X) (although Q(X) does not, in general, become a topological ring). Let - denote the metric completion and let * denote bounded functions. Then: C(X) is a ring of quotients of C*(X), so that Q(X) = Q(bX); Q-*(X) is a topological ring; Q-*(X) = Q*-(X); and Q-(X) is the maximal ring of quotients of Q-*(X). (bX denotes the Stone-Cech compactification of X).

The ring Q-(X) may be realized as the ring of all continuous functions on the dense Gd-sets in X (modulo an equivalence relation).

Let K denote the maximal ideal space of Q(X). The space Kis compact and extremally disconnected and is homeomorphic with the maximal ideal spaces of Q*(X), of Q-(X), and of Q-*(X). In addition, K is homeomorphic with the maximal ideal space of E(Q(X)), where E(A) denotes the Boolean ring of all idempotents of a ring A. Incidentally, E(Q-(X) = E(Q(X)); this ring is also isomorphic with the Boolean algebra B(X) of all regular open subsets of X , so that K is homeomorphic with the maximal ideal space of B(X). (It follows that K is the same for all separable metric spaces X without isolated points.) Finally, K may be realized as the inverse limit of the spaces bU, U ranging over all dense opens sets in X; the spaces bV with V ranging over all dense open sets in bX; and the spaces bS with S ranging over all dense Gd's in bX.

Results obtained in the course of the study include new proofs of known theorems, notably, the Stone-Nakano Theorem that C(X) is conditionally complete as a lattice if and only if X is extemally disconnected, and Artin's Theorem that in a formally real field, any element that is positive in every total order is a sum of squares.

### Editors' Preface

It has been forty years since these notes were printed (and forty-five since the results were announced). The results have been widely used, even though the original printing was not widely advertised and has long since been exhausted. Thus it seemed like a good idea to reprint it and make it freely available. Professors Gillman and Lambek were very supportive and readily assented to this. Unfortunately Professor Fine died in 1995 and, despite some effort, we have been unable to locate either his widow or his daughter. On the other hand, there is no copyright notice on the original edition and it would not seem that there is any possibility of commercial exploitation so that we have decided to go ahead nonetheless.

Since the original is a typescript, and not even high quality typescript, it was clear from the beginning that we would have it retyped by mathematicians who volunteered their services for the good of the community. We heartily thank these volunteers: Walter Burgess, Igor Khavkine, Chawne M. Kimber, Michelle L. Knox, Suzanne Larson, Ronald Levy, Warren Wm. McGovern, Jay Shapiro, and Eric R. Zenk. We also thank Gordon Mason and R. Grant Woods who have contributed to the pro ject by proofreading the final result.

Michael Barr
Robert Raphael

Montreal, August, 2005

Topology Atlas Document #iaal-94. Received Aug. 25, 2005.