Directory UMM :Journals:Journal_of_mathematics:TAC:
Theory and Appli
ations of Categories,
Vol. 6, No. 8, 1999, pp. 94{104.
SOME EPIMORPHIC REGULAR CONTEXTS
Dedi
ated to Joa
him Lambek on the o
asion of his 75th birthday
R. RAPHAEL
ABSTRACT. A von Neumann regular extension of a semiprime ring naturally de nes
a epimorphi
extension in the
ategory of rings. These are studied, and four natural
examples are
onsidered, two in
ommutative ring theory, and two in rings of
ontinuous
fun
tions.
1. Introdu
tion
In this note we study
ertain extensions of
ommutative semiprime rings whi
h we
all
epimorphi
regular
ontexts, or sometimes simply
ontexts. Work by Olivier shows that
there exists a universal su
h
ontext for ea
h semiprime ring. It turns out that these
extensions are \tangible" (
f. Theorem 2.2 and Remark 3.4) and this fa
ilitates their
study in
on
rete situations. We present four instan
es of epimorphi
regular
ontexts.
The rst is the epimorphi
hull due to Storrer, a ring that is de ned by a universal property
whose formulation is
ategori
al. The se
ond is the minimal regular rings as studied by
Pier
e and Burgess. The last two are topologi
al in nature. As a detailed instan
e of
the rst example one
onsiders the epimorphi
hull of a ring of
ontinuous fun
tions as
studied in [19℄. Lastly one
onsiders a general
ontext (
urrently being studied in [7℄)
whi
h
an be de ned naturally on any topologi
al spa
e.
The theme of this note is that
ategori
al methods
an stimulate fruitful inquiry in
other mathemati
al dis
iplines|in our
ase ring theory and topology. In the studies we
ite, new notions were de ned and new examples were dis
overed. Sometimes the notions
are fully or partially \internal" to their dis
ipline. Yet it seems most unlikely that they
would have been un
overed without the \external" stimulus from
ategory theory.
There were several motivations for writing this note. There has been
onsiderable
re
ent work on epimorphisms and rings of quotients in topologi
al settings ([7℄, [19℄, [20℄,
[21℄, [22℄) and non-algebraists might pro t from an introdu
tion to some of the ringtheoreti
and
ategori
al aspe
ts. As well, some of the \folklore" material is in danger of
being lost if it is not re
orded. There is also the hope that others will be en
ouraged to
solve the open problems and to
ommuni
ate additional instan
es of epimorphi
regular
This work was supported by the NSERC of Canada. The author is indebted to the referee for his
suggestions.
Re
eived by the editors 1999 January 22 and, in revised form, 1999 August 26.
Published on 1999 November 30.
1991 Mathemati
s Subje
t Classi
ation: 13A99, 18A20, 16B50.
Key words and phrases: epimorphism, von Neumann regular.
R. Raphael 1999. Permission to
opy for private use granted.
94
Theory and Appli
ations of Categories, Vol. 6, No. 8
95
ontexts among the rings that arise naturally in mathemati
s. Most of the open questions
have been
onsidered either with Storrer or in the
ollaborations with Henriksen, Ma
oosh,
and Woods. I am indebted to all of them for many stimulating
onversations in algebra
and in topology.
2. Epimorphisms
!
Re
all that a map f : R
S is
alled an epimorphism if g and h must
oin
ide if they
are maps from S
T whose
ompositions with f agree. A good introdu
tion to general
epimorphisms is found in [1℄. In
on
rete
ategories, su
h as
ategories of rings, an onto
map is an epimorphism, but so is the in
lusion of Z into Q. There is a substantial
literature on ring epimorphisms with many results of interest. To mention just two|
an epimorphi
extension of a
ommutative ring must be
ommutative [24, 1.3℄, and an
epimorphi
extension of a nite ring need not be nite [10, p. 268℄.
We are interested in the
ase of epimorphisms between rings whi
h are
ommutative
with identity, and semiprime, meaning that 0 is the only nilpotent element. Regular rings
are pertinent to the study of epimorphisms as we shall presently see. A ring R is
alled
regular in the sense of von Neumann if for all a R there is a b R su
h that a = aba.
The element b is
alled a quasi-inverse for a. As pointed out in [11, p. 36 Ex. 3℄ ea
h
element a has a unique quasi-inverse a that also satis es a = a aa .
!
2
2
2.1. Von Neumann Regular extensions and the universal regular ring of
Suppose that R is a semiprime subring of a general
ommutative regular ring
. Let G(S ) be the interse
tion of the
ommutative regular subrings of S that
ontain R.
From the existen
e of the unique quasi-inverses in the
ommutative
ase, it is easy to see
that G(S ) is itself a regular ring, and it is
learly the smallest regular subring of S that
ontains R. What is mu
h less evident, and
ru
ial to our studies, is the following:
Olivier.
S
[Olivier, [15℄℄
(i) The ring G(S ) is generated by R and the unique quasi-inverses of the elements
of R in S ; i.e. ea
h element of G(S ) is a nite sum of produ
ts of the form
d where
; d
R, and * denotes the unique quasi-inverse.
(ii) The embedding of R into G(S ) is an epimorphism of rings.
2.2. Theorem.
2
One attra
tive way of seeing (ii) on
e one has (i) is to note that ea
h d satis es a
simple form of the zig-zag
ondition due to Isbell [23, 3.3℄. Part (i) of the theorem follows
from the following key result.
[Olivier, [15℄℄ Given a
ommutative ring R, there is a universal obje
t
T (R) among epimorphi
regular extensions of R. The ring T (R) maps (over R)
anoni
ally onto any epimorphi
regular extension of R.
2.3. Theorem.
The fun
tor T is the left adjoint to the in
lusion of the
ategory of von
Neumann regular rings into the
ategory of
ommutative rings. Olivier's
onstru
tion
of T (R) is built by adjoining to R one indeterminate for ea
h element of R, and then
2.4. Remark.
Theory and Appli
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96
dividing out the minimal relations needed to ensure regularity for the elements of R. A
lo
alization argument shows that the ring T (R) is itself regular [15℄. Ea
h element of
T (R) is nitely expressible in terms of the elements of R and their quasi-inverses. Sin
e
T (R) maps onto G(S ) over R, one has part (i) of Theorem 2.2. Noti
e that it also shows:
If R is an in nite
ommutative semiprime ring and
phi
regular extension of R, then R and S have the same
ardinality.
2.5. Corollary.
S
is an epimor-
By an epimorphi
regular
ontext we will mean a map that is both a
monomorphism and an epimorphism from a
ommutative semiprime ring into a regular
ring. Note that there is no suggestion that the monomorphism is regular in the (
ategori
al) sense of being a
oequalizer.
2.6. Definition.
It is
lear from the above dis
ussion that any extension of a semiprime ring by a
regular ring gives rise to an epimorphi
regular
ontext. We shall presently examine
several natural
ases of this phenomenon.
Let us return to the fa
t that ea
h
element in G(S ) has a representation of the form indi
ated in Theorem 2.2. Adopting the
terminology of [7℄ let us de ne the regularity degree of an element x of G(S ) with respe
t
to R to be the least number of terms of the form
d ;
; d R, needed to represent x.
The regularity degree of G(S ) is the supremum of the regularity degrees of its elements.
Having regularity degree 0 means that R itself is regular. Having regularity degree in nite
means that there is no supremum for the regularity degrees of the individual elements of
R.
There are some natural questions whi
h arise:
2.7. The question of regularity degree.
2
!
Can one have an epimorphi
regular extension R
S of in nite regularity degree, or must every su
h extension be of nite regularity degree?
2.8. Problem.
Suppose that every epimorphi
regular extension is of nite degree. Are
there su
h extensions of arbitrary large regularity degree, or is there a global bound for
these degrees as one ranges over all epimorphi
regular extensions of semiprime rings? In
view of Theorem 2.3 it suÆ
es to determine the regularity degrees for Olivier's extensions
R
T (R).
2.9. Problem.
!
Problem 2.9 is
losely related to the following:
2.10. Problem. Are dire
t produ
ts of epis emanating from
ommutative rings, still
epi? Stated pre
isely if Ri ; Si are families of rings so that ea
h Si is an epimorphi
extension of Ri , is Si an epimorphi
extension of Ri ? The answer is known to be
negative in the non-
ommutative
ase [24, 2℄.
Q
Q
It is easy to see there are impli
ations between Problems 2.8, 2.9, and 2.10. In view of
Theorems 2.2(i) and 2.3 the existen
e of a global bound on regularity degrees implies that
dire
t produ
ts of epimorphi
regular extensions are epis. And if there are epimorphi
regular extensions of unbounded degree then Problem 2.10 has a negative answer by an
easy argument using dire
t produ
ts and Theorem 2.2(i) above. For the same reason a
Theory and Appli
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97
positive reply to Problem 2.8 means a negative answer to Problem 2.10. From re
ent
joint work with Henriksen and Woods it appears that there are examples using rings of
fun
tions for whi
h the degrees are nite, but unbounded. This predi
ts a negative answer
to Problem 2.10. There remains the question of getting \algebrai
" examples as answers
to these problems.
In studying the
ontext R ! G(S ) determined by a parti
ular regular
extension R ! S one has
onsiderable ma
hinery at one's disposal. First of all Olivier's
universal ring is
lose-by, as is the the epimorphi
hull, another universal obje
t, due to
Storrer, whi
h will be des
ribed presently. Se
ondly, regular rings have many attra
tive
properties in their own right. Thirdly, epimorphisms impose added stru
ture, for instan
e,
a result due to Lazard [14, 1.6℄ says that the
ontra
tion map Spe
G(S ) ! Spe
(R) must
be one-to-one. Lastly, one has a \handle" on the elements of G(S ) in that they have the
nite representation mentioned in Theorem 2.2(i). Although this representation is not
unique, its very existen
e is useful.
2.11. Overview.
3. Examples of epimorphi
regular
ontexts
There are two possible ways of nding a
ontext. It may be that we are given both a
semiprime ring R and a regular overring S , from whi
h we build G(S ). Or it maybe
that S itself is built by a natural
onstru
tion that begins with R. We shall en
ounter
examples of both sorts. Our rst examples are algebrai
.
To
de ne this ring we need another
ategori
al notion. A monomorphism f A ! B is
alled essential if ea
h map g : B ! C must be a monomorphism whenever g Æ f is a
monomorphism. Usually one dis
usses essentiality for extensions of modules but the
de nition makes sense in any
ategory (
f [23℄).
An epimorphi
hull E (A) for an obje
t A is an essential monoepi f : A ! E (A) with
the property that if h: A ! B is an essential monoepi, there is a map g : B ! E (A) so that
g Æ h = f . Storrer [23℄ showed that epimorphi
hulls are unique up to isomorphism and
studied the epimorphi
hulls of
ommutative semiprime rings in detail. Let us re
all that
if R is a semiprime ring and S is an extension of R, then S is
alled a ring of quotients of
R if given s 6= 0 2 R, there exists a t 2 S so that st 2 R; st 6= 0. The simplest example
of a ring of quotients is the passage from Z to Q, or, indeed, from any domain to its eld
of fra
tions. Rings of quotients have been studied in many ways and from many points
of view. A ring of quotients is an essential extension in the
ategory of rings. Every
semiprime ring R has a maximal (or \
omplete") ring of quotients denoted Q(R) whi
h
is unique up to isomorphism over R [11, 2.3℄. It is a regular ring [11, 2.4℄ so one has the
following:
3.1. The epimorphi
hull of a
ommutative semiprime ring [Storrer℄.
3.2. Theorem.
[[23℄℄
Let R be a
ommutative semiprime ring. Then R has an epimor-
( )
phi
hull in the
ategory of rings, denoted E R . It is a ring of quotients of R and is the
Theory and Appli
ations of Categories, Vol. 6, No. 8
98
epimorphi
regular
ontext determined by R and its
omplete ring of quotients Q(R). It
is (up to isomorphism) the only epimorphi
regular extension whi
h is essential.
Thus we have two important rings asso
iated with a general
ommutative semiprime
ring R, | T (R), due to Olivier, and E (R), due to Storrer. Ea
h has a universal property.
The two rings are related as follows:
[Folklore℄ Let R be a
ommutative semiprime ring. Let T (R) be its universal epimorphi
regular extension, and E (R) be its epimorphi
hull. There is a homomorphism from T (R) onto E (R) that xes R. (Indeed any epimorphi
regular extension
of R maps onto E (R) over R.)
3.3. Theorem.
For one thing, su
h a map must exist by the universal property of T (R). But in
fa
t, the ni
er way to see this is to use Zorn's lemma to
hoose in T (R) an ideal I maximal
with respe
t to having (0) interse
tion with R. Although the ideal I is far from unique,
the ring T (R)=I is a regular essential mono-epi, and hen
e a
opy of E (R). Note that
there are
ertain topologi
al situations when the
hoi
e of the ideal I is \
anoni
al".
Proof.
I am indebted to Hans Storrer for pointing out to me in private
orresponden
e the following fas
inating point
on
erning E (S ). Sin
e r = r, and (rs) = r s
one
an establish
part (i) of Theorem 2.2 for E (S ) if one knows an expression for a
P
n
quasi-inverse of i=1 ri si and this is given by
3.4. Remark.
n
X
i=1
! X Y
ri si
=
M i2M
(1
0 0
Y
X Y
si si )
sj
j2
=M
k2M
=
1 1
sl A rk A
l2
= (M[fkg)
where M runs through all subsets of f1; 2; : : : ; ng.
In order to present this example,
we must re
all some notions from studies by Pier
e [17℄, Burgess [1℄, and Burgess and
Stewart [3℄.
Suppose that S is a ring. The
hara
teristi
subring of S , denoted (S ), is the maximal
epimorphi
extension of the
anoni
al image of Z in S . By B (S ) one denotes the algebra
of
entral idempotents of S . The minimal subring of S (denoted P (S ) in [1℄) is the
subring of S that is generated by (S ) and B (S ). When P (S ) = S , S is
alled a minimal
ring. The minimal subring of a regular ring is itself regular. A regular ring is minimal,
exa
tly when its eld images are primitive elds. Minimal regular rings were studied by
Pier
e [17℄ who showed that they form a
ategory that is
ontravariantly equivalent to
the
ategory of boolean spa
es over the one-point
ompa
ti
ation of the set of primes
alled the
ategory of \labelled Boolean spa
es" (LBS). The
ategory of minimal regular
rings has some interesting properties, for example, it is
omplete and
o-
omplete.
To relate these notions to epimorphi
regular
ontexts we suppose that the ring S is
regular, and that R is the subring generated by B (S ) and the image of Z in S . Then one
has the following:
3.5. Minimal regular rings (Pier
e, Burgess).
Theory and Appli
ations of Categories, Vol. 6, No. 8
3.6. Proposition. (i)
(ii)
S
P (S ) = G(S ),
is the
lassi
al ring of quotients of
99
R
and hen
e the regularity degree of
G(S )
R is one.
Proof. (i) Sin
e G(S ) is the smallest regular ring between R and S , and P (S ) is regular,
one knows that G(S ) is a subring of P (S ). On the other hand G(S ) and P (S ) have the
same idempotents, so when these regular rings are represented as sheaves over Boolean
spa
es using the Pier
e sheaf [16℄ they have a
ommon base spa
e X = Spe
B (S ). For
ea
h x 2 X the stalks G(S )x and P (S )x are elds, and G(S )x is a sub eld of P (S )x. Sin
e
the stalks of P (S ) are primitive elds, the stalks of the two rings
oin
ide, as must the
rings themselves.
(ii) Suppose that r 2 R. By the regularity of S , the annihilator of r in S has the form
eS for some idempotent e of S . Sin
e B (S ) = B (R), the annihilator of r in R is eR and
it is easy to
he
k that this makes Q
l (R) regular. Furthermore ea
h non zero-divisor
in R is a non zero-divisor, and hen
e invertible in S . The universal property of Q
l (R)
implies that it maps into S over R. Sin
e its image is a regular ring between R and G(S ),
it is G(S ). The kernel of the map is
learly trivial, establishing the isomorphism. Sin
e
the elements in Q
l(R) have the form ab 1 ; a; b 2 R, the regularity degree of G(S ) with
respe
t to R is one.
3.7. Remark. It follows that one
annot build a
ounterexample to Problem 2.10 using
the rings of Proposition 3.6.
3.8. Rings of
ontinuous fun
tions. One is led naturally to
onsider rings of the
form C (X ), the algebra of all
ontinuous fun
tions on a (
ompletely regular) Hausdor
spa
e. There are two reasons for this: rst, if the algebrai
notions are valid, they may
have interpretations of interest in rings of fun
tions; and se
ond, rings of fun
tions are
suÆ
iently
ompli
ated that they are likely to provide examples and
ounterexamples for
algebrai
questions. This was already
lear when Storrer studied the epimorphi
hull [23,
11.6℄
Let us rst re
all some basi
notions, bearing in mind that [5℄ is the standard referen
e
for the topi
. If X is a topologi
al set and f : X ! R is a
ontinuous fun
tion, then the
zero-set of f is fx 2 X j f (x) = 0g. A subset of X is a GÆ if it is a
ountable interse
tion
of open sets. A point p 2 X is
alled a P -point of X if the zero-set of f is a neighbourhood
of p whenever f 2 C (X ) and f (p) = 0.
A topologi
al spa
e X is
alled
ompletely regular [5℄ if it is a Hausdor spa
e in whi
h
any point and any
losed set disjoint from it
an be
ompletely separated by a
ontinuous
real-valued fun
tion on X . When
onsidering the ring of all
ontinuous fun
tions on a
topologi
al spa
e one
an assume, without loss of generality that X is
ompletely regular
[5, Chapter 3℄.
3.9. -algebras. There are several simultaneous stru
tures on C (X )|that of a
ommutative algebra over R , that of a partially ordered set, and that of a latti
e. In [8℄ these
essential features were abstra
ted to the notion of a -algebra whose de nition requires
the following notions. Let A denote a latti
e ordered ring, and let A+ = fa 2 A: a 0g.
with respe
t to
Theory and Appli
ations of Categories, Vol. 6, No. 8
100
A latti
e-ordered ring A is
alled Ar
himedian if, for ea
h element a whi
h is di erent
from 0 the set fna: n 2 Z g has no upper bound in A. An element a 2 A+ is
alled a
weak order unit of A of if b 2 A and a _ b = 0 imply b = 0. Lastly, an ideal of A is
alled
an l-ideal, if a 2 I; b 2 A, and jbj jaj imply b 2 I . A -algebra is an Ar
himedean
latti
e-ordered algebra over R in whi
h 1 is a weak order unit. Ea
h C (X ) is a -algebra
but the
onverse is far from true. If A is a -algebra then M (A) denotes the set of maximal l-ideals in A endowed with the hull-kernel topology. For ea
h M 2 M (A), A=M is
a totally ordered eld
ontaining R . The ideal M is
alled real or hyper-real a
ordingly
as A=M
oin
ides with R or properly
ontains
it. The (possibly empty) subset of real
T
maximal l-ideals is denoted R(A), and if R(A) = (0), then A is a subdire
t produ
t of
opies of R and one
alls A a -algebra of real-valued fun
tions.
Henriksen and Johnson [8, 2.3℄ showed that any -algebra A has a representation as
an algebra of
ontinuous fun
tions to the extended reals de ned on the spa
e M (A). Ea
h
element of A is real on a dense open subset of M (A) and fun
tions are identi ed when
they agree on the interse
tion of the sets where they are real-valued. There is a natural
(supremum) metri
on A, so that it makes sense to speak of -algebras being uniformly
losed, (meaning that Cau
hy sequen
es
onverge).
A natural and re
urring question is whether a given -algebra is isomorphi
to a ring
of the form C (X ). Henriksen and Johnson [8, 5.2℄ found
onditions whi
h are ne
essary
and suÆ
ient|the ring must be a -algebra of real-valued fun
tions, it must be uniformly
losed, be
losed under inversion, and R(A) must be z -embedded in M (A) .
3.10. P -spa
es. A topologi
al spa
e is
alled a P -spa
e if every GÆ is open, equivalently,
if every zero-set is open. Compa
t P -spa
es are ne
essarily nite and
ountable P -spa
es
are dis
rete, but there exist P -spa
es with no isolated points. These spa
es are pertinent
be
ause X is a P -spa
e pre
isely when C (X ) is its own epimorphi
hull.
3.11. Almost P -spa
es. A spa
e X is almost P if every non-empty zero-set has a
non-empty interior, equivalently every nonempty GÆ set has non-empty interior. Unlike
P spa
es, almost P spa
es
an be
ompa
t and in nite, though they also are dis
rete
if they are
ountable. Watson [25℄ has
onstru
ted in ZFC a
ompa
t almost P spa
e
with no P -points. There is also an interesting spa
e due to Levy [12℄ who invented
these spa
es [13℄. On
e again there is an algebrai
hara
terization: it is pre
isely when
C (X ) = Q
l (X ) that X is almost P .
3.12. The delta topology. [18, 1W℄ There is a
anoni
al way of making a spa
e X
into a P -spa
e by (if ne
essary) strengthening its topology. One takes as a base for the
new topology all GÆ sets in the original spa
e. It is equivalent to take the set of zero-sets as
a base for a new topology. The new spa
e, denoted XÆ , is
alled the P -spa
e
ore
e
tion
of X . If j is the
ontinuous identity map XÆ ! X then one has the following universal
property: if Y is a Ty
hono P -spa
e and f : Y ! X is a
ontinuous map, then there is
a
ontinuous map k: Y ! XÆ so that j Æ k = f . In
ategori
al terms, P -spa
es form a
ore
e
tive sub
ategory of the
ategory of Ty
hono spa
es.
Theory and Appli
ations of Categories, Vol. 6, No. 8
101
The
lassi
al and
omplete rings of quotients
of a C (X ) were
onsidered as instan
es of the general algebrai
study by Fine, Gillman,
and Lambek in their now
lassi
[4℄. We will use their notation for these rings, Q
l (X ),
and Q(X ) respe
tively. The main result reads:
3.14. Theorem. [Fine, Gillman, Lambek, [4, 2.6℄℄ The ring Q(X ) is the set of all
ontinuous fun
tions de ned on dense open sets of X modulo the equivalen
e relation that
identi es two fun
tions that agree on the interse
tion of their domains. The ring Q
l (X )
is the algebra of all
ontinuous fun
tions on dense
ozero sets of X modulo the same
3.13. The epimorphi
hull of C(X).
equivalen
e relation.
Note that Q
l (X ) and Q(X ) are easily seen to be -algebras. Hager showed that
Q(X ) is rarely a ring of fun
tions as follows:
3.15. Theorem. [Hager [6℄℄
Suppose that the spe
trum of Q(X ) is of non-measurable
ardinality. Then Q(X ) is
isomorphi
to a C (Y ) iff X has a dense set of isolated points.
The epimorphi
hull of C (X ) was studied in [19℄ but it was denoted H (X ) be
ause
the symbol E (X ) had an existing and
on
i
ting meaning in topology. The epimorphi
hull was shown to be a -algebra for all spa
es X . The most natural (and still open)
question was:
3.16. Problem. Let X be a Ty
hono spa
e. Find ne
essary and suÆ
ient
onditions
on X for H (X ) to be isomorphi
to a C (Y ).
3.17. Subproblem. Find ne
essary and suÆ
ient
onditions on X for the -algebra
H (X ) to be uniformly
losed.
Problem 3.16 has never been solved. Indeed, one does not even know whether X must
ontain a P -point if H (X ) is a C (Y ), although this was re
ently established positively
for basi
ally dis
onne
ted spa
es [7℄. The presen
e of a dense set of P -points, indeed of
isolated points, does not suÆ
e to give that H (X ) is a C (Y ) [7℄.
Here is a summary of the main things one does know [19℄:
1. If H (X ) is a C (Y ), then
ompa
t subspa
es of X are s
attered.
2. If X has a
ountable dense set of isolated points then H (X ) = C (N ), where N
denotes a
ountable dis
rete spa
e.
3. One has a
hara
terization of when Q
l (X ) is a C (Y ).
4. One has many examples of spa
es, almost P and not, for whi
h H (X ) is a C (Y ),
as well as examples where some but not all of the three rings of quotients Q
l (X ), H (X )
and Q(X ) are isomorphi
to rings of
ontinuous fun
tions.
An important tool for studying H (X ) is the de nition of the following \test" subspa
e:
3.18. Definition. By gX denote the interse
tion of the dense
ozero sets of a real
ompa
t Ty
hono spa
e X .
One shows the following: let X be real
ompa
t and Ty
hono :
(i) The spa
e gX is a real
ompa
t almost P spa
e.
Theory and Appli
ations of Categories, Vol. 6, No. 8
102
(ii) If X has a dense almost P spa
e it has a largest one and it is gX .
(iii) The following are equivalent:
(a) H (X ) is a -algebra of real-valued fun
tions,
(b) gX is dense in X ,
(
) C (X ) has a regular ring of quotients of the form C (Y ).
This shows that (
f Problem 3.16):
(iv) If H (X ) is a C (Y ) then Y = (gX )Æ , (gX with the delta topology).
3.19. A
ontext of real-valued fun
tions. More re
ently there has been parallel
work undertaken by Henriksen, Raphael, and Woods whi
h drops the rings of quotients
aspe
t but keeps the dis
ussion very
on
rete. The idea is the following: Let X be a
Ty
hono spa
e and observe that C (X ) lies in the natural von Neumann regular ring
F (X ) of all real-valued fun
tions de ned on X . Let G(X ) be the
ontext determined by
C (X ) and F (X ). Like H (X ), the ring G(X ) is always a -algebra but unlike H (X ) it is
always a -algebra of real-valued fun
tions.
When X is an almost P spa
e, H (X ) and G(X )
oin
ide. In general these rings di er
but H (X ) is always a homomorphi
image of G(X ).
One is no longer dealing with rings of quotients, but still has very natural problems
su
h as:
3.20. Problem. Find ne
essary and suÆ
ient
onditions on X so that G(X ) be a ring
of
ontinuous fun
tions.
This is known to o
ur for some non P spa
es, but it o
urs so rarely that one raises
a more general question. Let Gu (X ) denote the set of limits of Cau
hy sequen
es from
G(X ).
3.21. Problem. Find ne
essary and suÆ
ient
onditions on X for Gu (X ) to be a ring.
[Of
ourse there is the parallel problem for H (X )℄. It is easy to see that Gu (X ) is
losed under sums, nite sups and infs, and s
alar multiples. Thus in the
ases where it
is a ring, it is a -algebra. It is also easy to see that being a ring is equivalent to being
losed under forming squares. Note that Isbell gave an example [9, 1.8℄ of an algebra of
real valued fun
tions whose uniform
losure is not
losed under taking squares.
Lastly returning to Olivier's work, there is the question
3.22. Problem. Des
ribe T (C (X )) where X is an arbitrary Ty
hono spa
e. Certainly
it has both G(X ) and H (X ) as a homomorphi
image. But when is it G(X )? When
is it a -algebra? When is it a -algebra of real-valued fun
tions or a uniformly
losed
-algebra?
Afterword Re
ent joint work with W.D. Burgess using algebrai
methods seems to give
progress on some of the problems presented above.
Theory and Appli
ations of Categories, Vol. 6, No. 8
103
Referen
es
[1℄ W.D. Burgess, The meaning of mono and epi in some familiar
ategories, Canad.
Math. Bull., 8, 1965, 759-769.
[2℄ W.D. Burgess, Minimal rings,
entral idempotents and the Pier
e sheaf, Contemp.
Math, 171, 1994, 51-67.
[3℄ W.D. Burgess and P.N. Stewart, The
hara
teristi
ring and the \best" way to
adjoin a one, J. Australian Math. So
iety, 47, 1989, 483-496.
[4℄ N.J. Fine, L. Gillman and J. Lambek, Rings of quotients of rings of fun
tions,
M
Gill University Press, Montreal, 1966.
[5℄ L. Gillman and M. Jerison, Rings of Continuous fun
tions, Van Nostrand, Prin
eton, 1960.
[6℄ A.W. Hager, Isomorphism with a C(Y) of the maximal ring of quotients of C(X),
Fund. Math. 66, 1969, 7-13.
[7℄ M. Henriksen, R. Raphael, R.G. Woods, Embedding a -algebra of real-valued fun
tions in a regular ring, in progress.
[8℄ M. Henriksen and D.G. Johnson, On the stru
ture of a
lass of ar
himedian latti
eordered algebras, Fund. Math. 50, 1961, 73-94.
[9℄ J.R. Isbell, Algebras of uniformly
ontinuous fun
tions, Ann. of Math. 68 (1958)
96-125.
[10℄ J.R. Isbell, Epimorphisms and dominions IV, J. London Math. So
., (2) I, 1969,
265-273.
[11℄ J. Lambek, Le
tures on Rings an Modules, Blaisdell, Toronto, 1966.
[12℄ R. Levy, Showering spa
es, Pa
i
Journal of Math. 57, 1975, 223-232.
[13℄ R. Levy, Almost P-spa
es, Can. J. Math, 29, 1977, 284-288.
[14℄ D. Lazard, Epimorphismes plats d'anneaux, C.R. A
ad. S
. Paris, 266 (5 fevrier
1968), Serie A, 314-316.
[15℄ J.P. Olivier, Anneaux absolument plats universels et epimorphismes d'anneaux,
C.R. A
ad. S
i. Paris, t. 266 Serie A (5 fevrier 1968), 317-318.
[16℄ R.S. Pier
e, Modules over
ommutative regular rings, Memoirs Amer. Math. So
.
70, Providen
e, 1967.
Theory and Appli
ations of Categories, Vol. 6, No. 8
104
[17℄ R.S. Pier
e, Minimal regular rings, Abelian groups and non-
ommutative rings,
Contemp. Math, 130, 335-348.
Extensions and absolutes of Hausdor spa
es,
[18℄ J.R. Porter and R.G. Woods,
Springer Verlag, 1988.
[19℄ R. Raphael and R.G. Woods,
and its Appli
ations.
The epimorphi
hull of C(X), to appear, Topology
f
[20℄ N. S
hwartz, Epimorphisms of -rings. Ordered algebrai
stru
tures (Cura
ao,
1988), 187{195, Math. Appl., 55, Kluwer A
ad. Publ., Dordre
ht, 1989.
[21℄ A.I. Singh and M.A. Swardson,
fun
tions, preprint, 1998.
[22℄ S.K. Jain and A.I. Singh,
preprint, 1998.
Quotient rings of algebras of fun
tions and operators,
[23℄ H.H. Storrer, Epimorphismen
43, 1968, 378-401.
[24℄ H.H.Storrer, Epimorphi
Helv., 48, 1973, 72-86.
Levels of quotient rings of rings of
ontinuous
von kommutativen Ringen, Comment. Math. Helv.,
extensions of Non-Commutative Rings, Comment. Math.
P
[25℄ Watson, S., A
ompa
t Hausdor spa
e without -points
interior, Pro
. Amer. Math. So
., 123, 1995, 2575-2577.
in whi
h
GÆ -sets have
Department of Mathemati
s,
Con
ordia University,
Montreal Canada
Email: raphaelal
or.
on
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ations of Categories,
Vol. 6, No. 8, 1999, pp. 94{104.
SOME EPIMORPHIC REGULAR CONTEXTS
Dedi
ated to Joa
him Lambek on the o
asion of his 75th birthday
R. RAPHAEL
ABSTRACT. A von Neumann regular extension of a semiprime ring naturally de nes
a epimorphi
extension in the
ategory of rings. These are studied, and four natural
examples are
onsidered, two in
ommutative ring theory, and two in rings of
ontinuous
fun
tions.
1. Introdu
tion
In this note we study
ertain extensions of
ommutative semiprime rings whi
h we
all
epimorphi
regular
ontexts, or sometimes simply
ontexts. Work by Olivier shows that
there exists a universal su
h
ontext for ea
h semiprime ring. It turns out that these
extensions are \tangible" (
f. Theorem 2.2 and Remark 3.4) and this fa
ilitates their
study in
on
rete situations. We present four instan
es of epimorphi
regular
ontexts.
The rst is the epimorphi
hull due to Storrer, a ring that is de ned by a universal property
whose formulation is
ategori
al. The se
ond is the minimal regular rings as studied by
Pier
e and Burgess. The last two are topologi
al in nature. As a detailed instan
e of
the rst example one
onsiders the epimorphi
hull of a ring of
ontinuous fun
tions as
studied in [19℄. Lastly one
onsiders a general
ontext (
urrently being studied in [7℄)
whi
h
an be de ned naturally on any topologi
al spa
e.
The theme of this note is that
ategori
al methods
an stimulate fruitful inquiry in
other mathemati
al dis
iplines|in our
ase ring theory and topology. In the studies we
ite, new notions were de ned and new examples were dis
overed. Sometimes the notions
are fully or partially \internal" to their dis
ipline. Yet it seems most unlikely that they
would have been un
overed without the \external" stimulus from
ategory theory.
There were several motivations for writing this note. There has been
onsiderable
re
ent work on epimorphisms and rings of quotients in topologi
al settings ([7℄, [19℄, [20℄,
[21℄, [22℄) and non-algebraists might pro t from an introdu
tion to some of the ringtheoreti
and
ategori
al aspe
ts. As well, some of the \folklore" material is in danger of
being lost if it is not re
orded. There is also the hope that others will be en
ouraged to
solve the open problems and to
ommuni
ate additional instan
es of epimorphi
regular
This work was supported by the NSERC of Canada. The author is indebted to the referee for his
suggestions.
Re
eived by the editors 1999 January 22 and, in revised form, 1999 August 26.
Published on 1999 November 30.
1991 Mathemati
s Subje
t Classi
ation: 13A99, 18A20, 16B50.
Key words and phrases: epimorphism, von Neumann regular.
R. Raphael 1999. Permission to
opy for private use granted.
94
Theory and Appli
ations of Categories, Vol. 6, No. 8
95
ontexts among the rings that arise naturally in mathemati
s. Most of the open questions
have been
onsidered either with Storrer or in the
ollaborations with Henriksen, Ma
oosh,
and Woods. I am indebted to all of them for many stimulating
onversations in algebra
and in topology.
2. Epimorphisms
!
Re
all that a map f : R
S is
alled an epimorphism if g and h must
oin
ide if they
are maps from S
T whose
ompositions with f agree. A good introdu
tion to general
epimorphisms is found in [1℄. In
on
rete
ategories, su
h as
ategories of rings, an onto
map is an epimorphism, but so is the in
lusion of Z into Q. There is a substantial
literature on ring epimorphisms with many results of interest. To mention just two|
an epimorphi
extension of a
ommutative ring must be
ommutative [24, 1.3℄, and an
epimorphi
extension of a nite ring need not be nite [10, p. 268℄.
We are interested in the
ase of epimorphisms between rings whi
h are
ommutative
with identity, and semiprime, meaning that 0 is the only nilpotent element. Regular rings
are pertinent to the study of epimorphisms as we shall presently see. A ring R is
alled
regular in the sense of von Neumann if for all a R there is a b R su
h that a = aba.
The element b is
alled a quasi-inverse for a. As pointed out in [11, p. 36 Ex. 3℄ ea
h
element a has a unique quasi-inverse a that also satis es a = a aa .
!
2
2
2.1. Von Neumann Regular extensions and the universal regular ring of
Suppose that R is a semiprime subring of a general
ommutative regular ring
. Let G(S ) be the interse
tion of the
ommutative regular subrings of S that
ontain R.
From the existen
e of the unique quasi-inverses in the
ommutative
ase, it is easy to see
that G(S ) is itself a regular ring, and it is
learly the smallest regular subring of S that
ontains R. What is mu
h less evident, and
ru
ial to our studies, is the following:
Olivier.
S
[Olivier, [15℄℄
(i) The ring G(S ) is generated by R and the unique quasi-inverses of the elements
of R in S ; i.e. ea
h element of G(S ) is a nite sum of produ
ts of the form
d where
; d
R, and * denotes the unique quasi-inverse.
(ii) The embedding of R into G(S ) is an epimorphism of rings.
2.2. Theorem.
2
One attra
tive way of seeing (ii) on
e one has (i) is to note that ea
h d satis es a
simple form of the zig-zag
ondition due to Isbell [23, 3.3℄. Part (i) of the theorem follows
from the following key result.
[Olivier, [15℄℄ Given a
ommutative ring R, there is a universal obje
t
T (R) among epimorphi
regular extensions of R. The ring T (R) maps (over R)
anoni
ally onto any epimorphi
regular extension of R.
2.3. Theorem.
The fun
tor T is the left adjoint to the in
lusion of the
ategory of von
Neumann regular rings into the
ategory of
ommutative rings. Olivier's
onstru
tion
of T (R) is built by adjoining to R one indeterminate for ea
h element of R, and then
2.4. Remark.
Theory and Appli
ations of Categories, Vol. 6, No. 8
96
dividing out the minimal relations needed to ensure regularity for the elements of R. A
lo
alization argument shows that the ring T (R) is itself regular [15℄. Ea
h element of
T (R) is nitely expressible in terms of the elements of R and their quasi-inverses. Sin
e
T (R) maps onto G(S ) over R, one has part (i) of Theorem 2.2. Noti
e that it also shows:
If R is an in nite
ommutative semiprime ring and
phi
regular extension of R, then R and S have the same
ardinality.
2.5. Corollary.
S
is an epimor-
By an epimorphi
regular
ontext we will mean a map that is both a
monomorphism and an epimorphism from a
ommutative semiprime ring into a regular
ring. Note that there is no suggestion that the monomorphism is regular in the (
ategori
al) sense of being a
oequalizer.
2.6. Definition.
It is
lear from the above dis
ussion that any extension of a semiprime ring by a
regular ring gives rise to an epimorphi
regular
ontext. We shall presently examine
several natural
ases of this phenomenon.
Let us return to the fa
t that ea
h
element in G(S ) has a representation of the form indi
ated in Theorem 2.2. Adopting the
terminology of [7℄ let us de ne the regularity degree of an element x of G(S ) with respe
t
to R to be the least number of terms of the form
d ;
; d R, needed to represent x.
The regularity degree of G(S ) is the supremum of the regularity degrees of its elements.
Having regularity degree 0 means that R itself is regular. Having regularity degree in nite
means that there is no supremum for the regularity degrees of the individual elements of
R.
There are some natural questions whi
h arise:
2.7. The question of regularity degree.
2
!
Can one have an epimorphi
regular extension R
S of in nite regularity degree, or must every su
h extension be of nite regularity degree?
2.8. Problem.
Suppose that every epimorphi
regular extension is of nite degree. Are
there su
h extensions of arbitrary large regularity degree, or is there a global bound for
these degrees as one ranges over all epimorphi
regular extensions of semiprime rings? In
view of Theorem 2.3 it suÆ
es to determine the regularity degrees for Olivier's extensions
R
T (R).
2.9. Problem.
!
Problem 2.9 is
losely related to the following:
2.10. Problem. Are dire
t produ
ts of epis emanating from
ommutative rings, still
epi? Stated pre
isely if Ri ; Si are families of rings so that ea
h Si is an epimorphi
extension of Ri , is Si an epimorphi
extension of Ri ? The answer is known to be
negative in the non-
ommutative
ase [24, 2℄.
Q
Q
It is easy to see there are impli
ations between Problems 2.8, 2.9, and 2.10. In view of
Theorems 2.2(i) and 2.3 the existen
e of a global bound on regularity degrees implies that
dire
t produ
ts of epimorphi
regular extensions are epis. And if there are epimorphi
regular extensions of unbounded degree then Problem 2.10 has a negative answer by an
easy argument using dire
t produ
ts and Theorem 2.2(i) above. For the same reason a
Theory and Appli
ations of Categories, Vol. 6, No. 8
97
positive reply to Problem 2.8 means a negative answer to Problem 2.10. From re
ent
joint work with Henriksen and Woods it appears that there are examples using rings of
fun
tions for whi
h the degrees are nite, but unbounded. This predi
ts a negative answer
to Problem 2.10. There remains the question of getting \algebrai
" examples as answers
to these problems.
In studying the
ontext R ! G(S ) determined by a parti
ular regular
extension R ! S one has
onsiderable ma
hinery at one's disposal. First of all Olivier's
universal ring is
lose-by, as is the the epimorphi
hull, another universal obje
t, due to
Storrer, whi
h will be des
ribed presently. Se
ondly, regular rings have many attra
tive
properties in their own right. Thirdly, epimorphisms impose added stru
ture, for instan
e,
a result due to Lazard [14, 1.6℄ says that the
ontra
tion map Spe
G(S ) ! Spe
(R) must
be one-to-one. Lastly, one has a \handle" on the elements of G(S ) in that they have the
nite representation mentioned in Theorem 2.2(i). Although this representation is not
unique, its very existen
e is useful.
2.11. Overview.
3. Examples of epimorphi
regular
ontexts
There are two possible ways of nding a
ontext. It may be that we are given both a
semiprime ring R and a regular overring S , from whi
h we build G(S ). Or it maybe
that S itself is built by a natural
onstru
tion that begins with R. We shall en
ounter
examples of both sorts. Our rst examples are algebrai
.
To
de ne this ring we need another
ategori
al notion. A monomorphism f A ! B is
alled essential if ea
h map g : B ! C must be a monomorphism whenever g Æ f is a
monomorphism. Usually one dis
usses essentiality for extensions of modules but the
de nition makes sense in any
ategory (
f [23℄).
An epimorphi
hull E (A) for an obje
t A is an essential monoepi f : A ! E (A) with
the property that if h: A ! B is an essential monoepi, there is a map g : B ! E (A) so that
g Æ h = f . Storrer [23℄ showed that epimorphi
hulls are unique up to isomorphism and
studied the epimorphi
hulls of
ommutative semiprime rings in detail. Let us re
all that
if R is a semiprime ring and S is an extension of R, then S is
alled a ring of quotients of
R if given s 6= 0 2 R, there exists a t 2 S so that st 2 R; st 6= 0. The simplest example
of a ring of quotients is the passage from Z to Q, or, indeed, from any domain to its eld
of fra
tions. Rings of quotients have been studied in many ways and from many points
of view. A ring of quotients is an essential extension in the
ategory of rings. Every
semiprime ring R has a maximal (or \
omplete") ring of quotients denoted Q(R) whi
h
is unique up to isomorphism over R [11, 2.3℄. It is a regular ring [11, 2.4℄ so one has the
following:
3.1. The epimorphi
hull of a
ommutative semiprime ring [Storrer℄.
3.2. Theorem.
[[23℄℄
Let R be a
ommutative semiprime ring. Then R has an epimor-
( )
phi
hull in the
ategory of rings, denoted E R . It is a ring of quotients of R and is the
Theory and Appli
ations of Categories, Vol. 6, No. 8
98
epimorphi
regular
ontext determined by R and its
omplete ring of quotients Q(R). It
is (up to isomorphism) the only epimorphi
regular extension whi
h is essential.
Thus we have two important rings asso
iated with a general
ommutative semiprime
ring R, | T (R), due to Olivier, and E (R), due to Storrer. Ea
h has a universal property.
The two rings are related as follows:
[Folklore℄ Let R be a
ommutative semiprime ring. Let T (R) be its universal epimorphi
regular extension, and E (R) be its epimorphi
hull. There is a homomorphism from T (R) onto E (R) that xes R. (Indeed any epimorphi
regular extension
of R maps onto E (R) over R.)
3.3. Theorem.
For one thing, su
h a map must exist by the universal property of T (R). But in
fa
t, the ni
er way to see this is to use Zorn's lemma to
hoose in T (R) an ideal I maximal
with respe
t to having (0) interse
tion with R. Although the ideal I is far from unique,
the ring T (R)=I is a regular essential mono-epi, and hen
e a
opy of E (R). Note that
there are
ertain topologi
al situations when the
hoi
e of the ideal I is \
anoni
al".
Proof.
I am indebted to Hans Storrer for pointing out to me in private
orresponden
e the following fas
inating point
on
erning E (S ). Sin
e r = r, and (rs) = r s
one
an establish
part (i) of Theorem 2.2 for E (S ) if one knows an expression for a
P
n
quasi-inverse of i=1 ri si and this is given by
3.4. Remark.
n
X
i=1
! X Y
ri si
=
M i2M
(1
0 0
Y
X Y
si si )
sj
j2
=M
k2M
=
1 1
sl A rk A
l2
= (M[fkg)
where M runs through all subsets of f1; 2; : : : ; ng.
In order to present this example,
we must re
all some notions from studies by Pier
e [17℄, Burgess [1℄, and Burgess and
Stewart [3℄.
Suppose that S is a ring. The
hara
teristi
subring of S , denoted (S ), is the maximal
epimorphi
extension of the
anoni
al image of Z in S . By B (S ) one denotes the algebra
of
entral idempotents of S . The minimal subring of S (denoted P (S ) in [1℄) is the
subring of S that is generated by (S ) and B (S ). When P (S ) = S , S is
alled a minimal
ring. The minimal subring of a regular ring is itself regular. A regular ring is minimal,
exa
tly when its eld images are primitive elds. Minimal regular rings were studied by
Pier
e [17℄ who showed that they form a
ategory that is
ontravariantly equivalent to
the
ategory of boolean spa
es over the one-point
ompa
ti
ation of the set of primes
alled the
ategory of \labelled Boolean spa
es" (LBS). The
ategory of minimal regular
rings has some interesting properties, for example, it is
omplete and
o-
omplete.
To relate these notions to epimorphi
regular
ontexts we suppose that the ring S is
regular, and that R is the subring generated by B (S ) and the image of Z in S . Then one
has the following:
3.5. Minimal regular rings (Pier
e, Burgess).
Theory and Appli
ations of Categories, Vol. 6, No. 8
3.6. Proposition. (i)
(ii)
S
P (S ) = G(S ),
is the
lassi
al ring of quotients of
99
R
and hen
e the regularity degree of
G(S )
R is one.
Proof. (i) Sin
e G(S ) is the smallest regular ring between R and S , and P (S ) is regular,
one knows that G(S ) is a subring of P (S ). On the other hand G(S ) and P (S ) have the
same idempotents, so when these regular rings are represented as sheaves over Boolean
spa
es using the Pier
e sheaf [16℄ they have a
ommon base spa
e X = Spe
B (S ). For
ea
h x 2 X the stalks G(S )x and P (S )x are elds, and G(S )x is a sub eld of P (S )x. Sin
e
the stalks of P (S ) are primitive elds, the stalks of the two rings
oin
ide, as must the
rings themselves.
(ii) Suppose that r 2 R. By the regularity of S , the annihilator of r in S has the form
eS for some idempotent e of S . Sin
e B (S ) = B (R), the annihilator of r in R is eR and
it is easy to
he
k that this makes Q
l (R) regular. Furthermore ea
h non zero-divisor
in R is a non zero-divisor, and hen
e invertible in S . The universal property of Q
l (R)
implies that it maps into S over R. Sin
e its image is a regular ring between R and G(S ),
it is G(S ). The kernel of the map is
learly trivial, establishing the isomorphism. Sin
e
the elements in Q
l(R) have the form ab 1 ; a; b 2 R, the regularity degree of G(S ) with
respe
t to R is one.
3.7. Remark. It follows that one
annot build a
ounterexample to Problem 2.10 using
the rings of Proposition 3.6.
3.8. Rings of
ontinuous fun
tions. One is led naturally to
onsider rings of the
form C (X ), the algebra of all
ontinuous fun
tions on a (
ompletely regular) Hausdor
spa
e. There are two reasons for this: rst, if the algebrai
notions are valid, they may
have interpretations of interest in rings of fun
tions; and se
ond, rings of fun
tions are
suÆ
iently
ompli
ated that they are likely to provide examples and
ounterexamples for
algebrai
questions. This was already
lear when Storrer studied the epimorphi
hull [23,
11.6℄
Let us rst re
all some basi
notions, bearing in mind that [5℄ is the standard referen
e
for the topi
. If X is a topologi
al set and f : X ! R is a
ontinuous fun
tion, then the
zero-set of f is fx 2 X j f (x) = 0g. A subset of X is a GÆ if it is a
ountable interse
tion
of open sets. A point p 2 X is
alled a P -point of X if the zero-set of f is a neighbourhood
of p whenever f 2 C (X ) and f (p) = 0.
A topologi
al spa
e X is
alled
ompletely regular [5℄ if it is a Hausdor spa
e in whi
h
any point and any
losed set disjoint from it
an be
ompletely separated by a
ontinuous
real-valued fun
tion on X . When
onsidering the ring of all
ontinuous fun
tions on a
topologi
al spa
e one
an assume, without loss of generality that X is
ompletely regular
[5, Chapter 3℄.
3.9. -algebras. There are several simultaneous stru
tures on C (X )|that of a
ommutative algebra over R , that of a partially ordered set, and that of a latti
e. In [8℄ these
essential features were abstra
ted to the notion of a -algebra whose de nition requires
the following notions. Let A denote a latti
e ordered ring, and let A+ = fa 2 A: a 0g.
with respe
t to
Theory and Appli
ations of Categories, Vol. 6, No. 8
100
A latti
e-ordered ring A is
alled Ar
himedian if, for ea
h element a whi
h is di erent
from 0 the set fna: n 2 Z g has no upper bound in A. An element a 2 A+ is
alled a
weak order unit of A of if b 2 A and a _ b = 0 imply b = 0. Lastly, an ideal of A is
alled
an l-ideal, if a 2 I; b 2 A, and jbj jaj imply b 2 I . A -algebra is an Ar
himedean
latti
e-ordered algebra over R in whi
h 1 is a weak order unit. Ea
h C (X ) is a -algebra
but the
onverse is far from true. If A is a -algebra then M (A) denotes the set of maximal l-ideals in A endowed with the hull-kernel topology. For ea
h M 2 M (A), A=M is
a totally ordered eld
ontaining R . The ideal M is
alled real or hyper-real a
ordingly
as A=M
oin
ides with R or properly
ontains
it. The (possibly empty) subset of real
T
maximal l-ideals is denoted R(A), and if R(A) = (0), then A is a subdire
t produ
t of
opies of R and one
alls A a -algebra of real-valued fun
tions.
Henriksen and Johnson [8, 2.3℄ showed that any -algebra A has a representation as
an algebra of
ontinuous fun
tions to the extended reals de ned on the spa
e M (A). Ea
h
element of A is real on a dense open subset of M (A) and fun
tions are identi ed when
they agree on the interse
tion of the sets where they are real-valued. There is a natural
(supremum) metri
on A, so that it makes sense to speak of -algebras being uniformly
losed, (meaning that Cau
hy sequen
es
onverge).
A natural and re
urring question is whether a given -algebra is isomorphi
to a ring
of the form C (X ). Henriksen and Johnson [8, 5.2℄ found
onditions whi
h are ne
essary
and suÆ
ient|the ring must be a -algebra of real-valued fun
tions, it must be uniformly
losed, be
losed under inversion, and R(A) must be z -embedded in M (A) .
3.10. P -spa
es. A topologi
al spa
e is
alled a P -spa
e if every GÆ is open, equivalently,
if every zero-set is open. Compa
t P -spa
es are ne
essarily nite and
ountable P -spa
es
are dis
rete, but there exist P -spa
es with no isolated points. These spa
es are pertinent
be
ause X is a P -spa
e pre
isely when C (X ) is its own epimorphi
hull.
3.11. Almost P -spa
es. A spa
e X is almost P if every non-empty zero-set has a
non-empty interior, equivalently every nonempty GÆ set has non-empty interior. Unlike
P spa
es, almost P spa
es
an be
ompa
t and in nite, though they also are dis
rete
if they are
ountable. Watson [25℄ has
onstru
ted in ZFC a
ompa
t almost P spa
e
with no P -points. There is also an interesting spa
e due to Levy [12℄ who invented
these spa
es [13℄. On
e again there is an algebrai
hara
terization: it is pre
isely when
C (X ) = Q
l (X ) that X is almost P .
3.12. The delta topology. [18, 1W℄ There is a
anoni
al way of making a spa
e X
into a P -spa
e by (if ne
essary) strengthening its topology. One takes as a base for the
new topology all GÆ sets in the original spa
e. It is equivalent to take the set of zero-sets as
a base for a new topology. The new spa
e, denoted XÆ , is
alled the P -spa
e
ore
e
tion
of X . If j is the
ontinuous identity map XÆ ! X then one has the following universal
property: if Y is a Ty
hono P -spa
e and f : Y ! X is a
ontinuous map, then there is
a
ontinuous map k: Y ! XÆ so that j Æ k = f . In
ategori
al terms, P -spa
es form a
ore
e
tive sub
ategory of the
ategory of Ty
hono spa
es.
Theory and Appli
ations of Categories, Vol. 6, No. 8
101
The
lassi
al and
omplete rings of quotients
of a C (X ) were
onsidered as instan
es of the general algebrai
study by Fine, Gillman,
and Lambek in their now
lassi
[4℄. We will use their notation for these rings, Q
l (X ),
and Q(X ) respe
tively. The main result reads:
3.14. Theorem. [Fine, Gillman, Lambek, [4, 2.6℄℄ The ring Q(X ) is the set of all
ontinuous fun
tions de ned on dense open sets of X modulo the equivalen
e relation that
identi es two fun
tions that agree on the interse
tion of their domains. The ring Q
l (X )
is the algebra of all
ontinuous fun
tions on dense
ozero sets of X modulo the same
3.13. The epimorphi
hull of C(X).
equivalen
e relation.
Note that Q
l (X ) and Q(X ) are easily seen to be -algebras. Hager showed that
Q(X ) is rarely a ring of fun
tions as follows:
3.15. Theorem. [Hager [6℄℄
Suppose that the spe
trum of Q(X ) is of non-measurable
ardinality. Then Q(X ) is
isomorphi
to a C (Y ) iff X has a dense set of isolated points.
The epimorphi
hull of C (X ) was studied in [19℄ but it was denoted H (X ) be
ause
the symbol E (X ) had an existing and
on
i
ting meaning in topology. The epimorphi
hull was shown to be a -algebra for all spa
es X . The most natural (and still open)
question was:
3.16. Problem. Let X be a Ty
hono spa
e. Find ne
essary and suÆ
ient
onditions
on X for H (X ) to be isomorphi
to a C (Y ).
3.17. Subproblem. Find ne
essary and suÆ
ient
onditions on X for the -algebra
H (X ) to be uniformly
losed.
Problem 3.16 has never been solved. Indeed, one does not even know whether X must
ontain a P -point if H (X ) is a C (Y ), although this was re
ently established positively
for basi
ally dis
onne
ted spa
es [7℄. The presen
e of a dense set of P -points, indeed of
isolated points, does not suÆ
e to give that H (X ) is a C (Y ) [7℄.
Here is a summary of the main things one does know [19℄:
1. If H (X ) is a C (Y ), then
ompa
t subspa
es of X are s
attered.
2. If X has a
ountable dense set of isolated points then H (X ) = C (N ), where N
denotes a
ountable dis
rete spa
e.
3. One has a
hara
terization of when Q
l (X ) is a C (Y ).
4. One has many examples of spa
es, almost P and not, for whi
h H (X ) is a C (Y ),
as well as examples where some but not all of the three rings of quotients Q
l (X ), H (X )
and Q(X ) are isomorphi
to rings of
ontinuous fun
tions.
An important tool for studying H (X ) is the de nition of the following \test" subspa
e:
3.18. Definition. By gX denote the interse
tion of the dense
ozero sets of a real
ompa
t Ty
hono spa
e X .
One shows the following: let X be real
ompa
t and Ty
hono :
(i) The spa
e gX is a real
ompa
t almost P spa
e.
Theory and Appli
ations of Categories, Vol. 6, No. 8
102
(ii) If X has a dense almost P spa
e it has a largest one and it is gX .
(iii) The following are equivalent:
(a) H (X ) is a -algebra of real-valued fun
tions,
(b) gX is dense in X ,
(
) C (X ) has a regular ring of quotients of the form C (Y ).
This shows that (
f Problem 3.16):
(iv) If H (X ) is a C (Y ) then Y = (gX )Æ , (gX with the delta topology).
3.19. A
ontext of real-valued fun
tions. More re
ently there has been parallel
work undertaken by Henriksen, Raphael, and Woods whi
h drops the rings of quotients
aspe
t but keeps the dis
ussion very
on
rete. The idea is the following: Let X be a
Ty
hono spa
e and observe that C (X ) lies in the natural von Neumann regular ring
F (X ) of all real-valued fun
tions de ned on X . Let G(X ) be the
ontext determined by
C (X ) and F (X ). Like H (X ), the ring G(X ) is always a -algebra but unlike H (X ) it is
always a -algebra of real-valued fun
tions.
When X is an almost P spa
e, H (X ) and G(X )
oin
ide. In general these rings di er
but H (X ) is always a homomorphi
image of G(X ).
One is no longer dealing with rings of quotients, but still has very natural problems
su
h as:
3.20. Problem. Find ne
essary and suÆ
ient
onditions on X so that G(X ) be a ring
of
ontinuous fun
tions.
This is known to o
ur for some non P spa
es, but it o
urs so rarely that one raises
a more general question. Let Gu (X ) denote the set of limits of Cau
hy sequen
es from
G(X ).
3.21. Problem. Find ne
essary and suÆ
ient
onditions on X for Gu (X ) to be a ring.
[Of
ourse there is the parallel problem for H (X )℄. It is easy to see that Gu (X ) is
losed under sums, nite sups and infs, and s
alar multiples. Thus in the
ases where it
is a ring, it is a -algebra. It is also easy to see that being a ring is equivalent to being
losed under forming squares. Note that Isbell gave an example [9, 1.8℄ of an algebra of
real valued fun
tions whose uniform
losure is not
losed under taking squares.
Lastly returning to Olivier's work, there is the question
3.22. Problem. Des
ribe T (C (X )) where X is an arbitrary Ty
hono spa
e. Certainly
it has both G(X ) and H (X ) as a homomorphi
image. But when is it G(X )? When
is it a -algebra? When is it a -algebra of real-valued fun
tions or a uniformly
losed
-algebra?
Afterword Re
ent joint work with W.D. Burgess using algebrai
methods seems to give
progress on some of the problems presented above.
Theory and Appli
ations of Categories, Vol. 6, No. 8
103
Referen
es
[1℄ W.D. Burgess, The meaning of mono and epi in some familiar
ategories, Canad.
Math. Bull., 8, 1965, 759-769.
[2℄ W.D. Burgess, Minimal rings,
entral idempotents and the Pier
e sheaf, Contemp.
Math, 171, 1994, 51-67.
[3℄ W.D. Burgess and P.N. Stewart, The
hara
teristi
ring and the \best" way to
adjoin a one, J. Australian Math. So
iety, 47, 1989, 483-496.
[4℄ N.J. Fine, L. Gillman and J. Lambek, Rings of quotients of rings of fun
tions,
M
Gill University Press, Montreal, 1966.
[5℄ L. Gillman and M. Jerison, Rings of Continuous fun
tions, Van Nostrand, Prin
eton, 1960.
[6℄ A.W. Hager, Isomorphism with a C(Y) of the maximal ring of quotients of C(X),
Fund. Math. 66, 1969, 7-13.
[7℄ M. Henriksen, R. Raphael, R.G. Woods, Embedding a -algebra of real-valued fun
tions in a regular ring, in progress.
[8℄ M. Henriksen and D.G. Johnson, On the stru
ture of a
lass of ar
himedian latti
eordered algebras, Fund. Math. 50, 1961, 73-94.
[9℄ J.R. Isbell, Algebras of uniformly
ontinuous fun
tions, Ann. of Math. 68 (1958)
96-125.
[10℄ J.R. Isbell, Epimorphisms and dominions IV, J. London Math. So
., (2) I, 1969,
265-273.
[11℄ J. Lambek, Le
tures on Rings an Modules, Blaisdell, Toronto, 1966.
[12℄ R. Levy, Showering spa
es, Pa
i
Journal of Math. 57, 1975, 223-232.
[13℄ R. Levy, Almost P-spa
es, Can. J. Math, 29, 1977, 284-288.
[14℄ D. Lazard, Epimorphismes plats d'anneaux, C.R. A
ad. S
. Paris, 266 (5 fevrier
1968), Serie A, 314-316.
[15℄ J.P. Olivier, Anneaux absolument plats universels et epimorphismes d'anneaux,
C.R. A
ad. S
i. Paris, t. 266 Serie A (5 fevrier 1968), 317-318.
[16℄ R.S. Pier
e, Modules over
ommutative regular rings, Memoirs Amer. Math. So
.
70, Providen
e, 1967.
Theory and Appli
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[17℄ R.S. Pier
e, Minimal regular rings, Abelian groups and non-
ommutative rings,
Contemp. Math, 130, 335-348.
Extensions and absolutes of Hausdor spa
es,
[18℄ J.R. Porter and R.G. Woods,
Springer Verlag, 1988.
[19℄ R. Raphael and R.G. Woods,
and its Appli
ations.
The epimorphi
hull of C(X), to appear, Topology
f
[20℄ N. S
hwartz, Epimorphisms of -rings. Ordered algebrai
stru
tures (Cura
ao,
1988), 187{195, Math. Appl., 55, Kluwer A
ad. Publ., Dordre
ht, 1989.
[21℄ A.I. Singh and M.A. Swardson,
fun
tions, preprint, 1998.
[22℄ S.K. Jain and A.I. Singh,
preprint, 1998.
Quotient rings of algebras of fun
tions and operators,
[23℄ H.H. Storrer, Epimorphismen
43, 1968, 378-401.
[24℄ H.H.Storrer, Epimorphi
Helv., 48, 1973, 72-86.
Levels of quotient rings of rings of
ontinuous
von kommutativen Ringen, Comment. Math. Helv.,
extensions of Non-Commutative Rings, Comment. Math.
P
[25℄ Watson, S., A
ompa
t Hausdor spa
e without -points
interior, Pro
. Amer. Math. So
., 123, 1995, 2575-2577.
in whi
h
GÆ -sets have
Department of Mathemati
s,
Con
ordia University,
Montreal Canada
Email: raphaelal
or.
on
ordia.
a
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.edu
Ross Street, Ma
quarie University: streetmath.mq.edu.au
Walter Tholen, York University: tholenmathstat.yorku.
a
Myles Tierney, Rutgers University: tierneymath.rutgers.edu
Robert F. C. Walters, University of Sydney: walters bmaths.usyd.edu.au
R. J. Wood, Dalhousie University: rjwoodms
s.dal.
a