SOME EPIMORPHIC REGULAR CONTEXTS
Dediated to Joahim Lambek on the oasion of his 75th birthday
R.RAPHAEL
ABSTRACT. AvonNeumannregularextensionofasemiprimeringnaturallydenes
a epimorphi extension in the ategory of rings. These are studied, and four natural
examplesareonsidered,twoinommutativeringtheory,andtwoinringsofontinuous
funtions.
1. Introdution
In this note we study ertain extensions of ommutative semiprime rings whih we all
epimorphi regular ontexts, or sometimes simplyontexts. Work by Oliviershows that
there exists a universal suh ontext for eah semiprime ring. It turns out that these
extensions are \tangible" (f. Theorem 2.2 and Remark 3.4) and this failitates their
study in onrete situations. We present four instanes of epimorphi regular ontexts.
TherstistheepimorphihullduetoStorrer,aringthatisdenedbyauniversalproperty
whose formulation is ategorial. The seond is the minimal regular rings as studied by
Piere and Burgess. The last two are topologial in nature. As a detailed instane of
the rst example one onsiders the epimorphi hull of a ring of ontinuous funtions as
studied in [19℄. Lastly one onsiders a general ontext (urrently being studied in [7℄)
whihan bedened naturallyon any topologialspae.
The theme of this note is that ategorial methods an stimulate fruitful inquiry in
other mathematial disiplines|in our ase ring theory and topology. In the studies we
ite,new notionswere denedand newexampleswere disovered. Sometimesthe notions
are fully or partially \internal" to their disipline. Yet it seems most unlikely that they
would have been unovered withoutthe \external" stimulus fromategory theory.
There were several motivations for writing this note. There has been onsiderable
reent workonepimorphisms andrings of quotientsin topologialsettings([7℄,[19℄, [20℄,
[21℄, [22℄) and non-algebraists might prot from an introdution to some of the
ring-theoreti and ategorial aspets. As well, someof the \folklore"materialis indangerof
being lostif it is not reorded. There is also the hope that others willbe enouraged to
solve the open problems and to ommuniateadditional instanes of epimorphi regular
This work was supported bythe NSERC of Canada. Theauthor is indebted to the refereefor his
suggestions.
Reeivedbytheeditors1999January22and, inrevisedform,1999August26.
Publishedon1999November30.
1991MathematisSubjetClassiation: 13A99,18A20,16B50.
Keywordsandphrases: epimorphism,vonNeumannregular.
ontextsamongtheringsthatarisenaturallyinmathematis. Mostoftheopen questions
havebeenonsideredeitherwithStorrerorintheollaborationswithHenriksen,Maoosh,
and Woods. I am indebted to all of them for many stimulating onversations in algebra
and in topology.
2. Epimorphisms
Reall that a map f:R ! S is alled an epimorphism if g and h must oinide if they
are maps from S !T whose ompositionswith f agree. A good introdutionto general
epimorphisms is found in [1℄. In onrete ategories, suh as ategories of rings, an onto
map is an epimorphism, but so is the inlusion 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 extensionof a nite ring need not benite [10, p.268℄.
We are interested in the ase of epimorphisms between rings whih are ommutative
withidentity,andsemiprime,meaningthat0istheonlynilpotentelement. Regularrings
are pertinent to the study of epimorphismsas we shall presently see. A ring R is alled
regular inthe sense of von Neumann if for all a 2R there is a b 2R suh that a =aba.
The element b is alled a quasi-inverse for a. As pointed out in [11, p. 36 Ex. 3℄ eah
element a has aunique quasi-inverse a
that also satisesa
=a
aa
.
2.1. Von Neumann Regular extensions and the universal regular ring of
Olivier. Suppose that R is a semiprime subring of a generalommutativeregular ring
S. LetG(S)betheintersetionof theommutativeregularsubringsofS that ontain R .
From the existeneof the unique quasi-inverses inthe ommutativease, itiseasy tosee
that G(S) is itself a regular ring, and it is learly the smallest regular subring of S that
ontains R . What is muh less evident,and ruialto our studies,is the following:
2.2. Theorem. [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. eah element of G(S) is a nite sum of produts of the form d
where
;d2R , and * denotes the unique quasi-inverse.
(ii) The embedding of R into G(S) is an epimorphism of rings.
One attrative way of seeing (ii) one one has (i) is to note that eah d
satises a
simpleformofthezig-zagonditiondue toIsbell[23,3.3℄. Part (i)of thetheorem follows
fromthe following key result.
2.3. Theorem. [Olivier, [15℄℄ Given a ommutative ring R , there is a universal objet
T(R ) among epimorphi regular extensions of R . The ring T(R ) maps (over R )
anoni-ally onto any epimorphi regular extension of R .
2.4. Remark. The funtor T is the left adjoint to the inlusion of the ategory of von
dividing out the minimal relations needed to ensure regularity for the elements of R . A
loalization argument shows that the ring T(R ) is itself regular [15℄. Eah element of
T(R ) is nitely expressible in terms of the elements of R and their quasi-inverses. Sine
T(R )maps ontoG(S)overR , onehas part (i)of Theorem 2.2. Notie thatitalsoshows:
2.5. Corollary. If R is an innite ommutative semiprime ring and S is an
epimor-phi regular extension of R , then R and S have the same ardinality.
2.6. Definition. 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 themonomorphism is regular in the
(ategori-al) sense of beinga oequalizer.
It is lear from the above disussion that any extension of a semiprime ring by a
regular ring gives rise to an epimorphi regular ontext. We shall presently examine
several naturalases of this phenomenon.
2.7. The question of regularity degree. Let us return to the fat that eah
elementinG(S)has arepresentation oftheformindiatedinTheorem2.2. Adoptingthe
terminologyof[7℄letusdenetheregularitydegreeofanelementx ofG(S)withrespet
to R to be the least number of terms of the form d
;;d 2 R , needed to represent x.
The regularity degree of G(S) is the supremum of the regularity degrees of its elements.
Havingregularitydegree0meansthatRitselfisregular. Havingregularitydegreeinnite
means that thereis nosupremum for the regularity degrees of the individual elementsof
R .
There are some natural questions whih arise:
2.8. Problem. Can one have an epimorphi regular extension R ! S of innite
regu-larity degree, or must every suh extension be of nite regularity degree?
2.9. Problem. Suppose thatevery epimorphiregularextensionisofnite degree. Are
there suh extensions of arbitrary large regularity degree, or is there a global bound for
these degrees asone rangesoverallepimorphi regularextensions of semiprimerings? In
viewofTheorem2.3itsuÆes todeterminethe regularitydegrees forOlivier'sextensions
R !T(R ).
Problem 2.9is losely relatedto the following:
2.10. Problem. Are diret produts of epis emanating from ommutative rings, still
epi? Stated preisely if R
i ; S
i
are families of rings so that eah S
i
is an epimorphi
extension of R
i , is
Q
S
i
an epimorphi extension of Q
R
i
? The answer is known to be
negativein the non-ommutativease [24, 2℄.
It iseasytosee thereare impliationsbetween Problems2.8,2.9, and2.10. Inviewof
Theorems2.2(i)and2.3theexisteneofaglobalboundonregularitydegrees impliesthat
diret produts of epimorphi regular extensions are epis. And if there are epimorphi
positive reply to Problem 2.8 means a negative answer to Problem 2.10. From reent
joint work with Henriksen and Woods it appears that there are examples using rings of
funtionsforwhihthedegreesarenite,butunbounded. Thispreditsanegativeanswer
toProblem 2.10. There remainsthe questionof getting\algebrai" examples asanswers
tothese problems.
2.11. Overview. InstudyingtheontextR !G(S)determinedbyapartiularregular
extension R !S one has onsiderablemahinery atone's disposal. First of allOlivier's
universal ring is lose-by, as is the the epimorphi hull, another universal objet, due to
Storrer, whih will be desribed presently. Seondly, regular rings have many attrative
propertiesintheirownright. Thirdly,epimorphismsimposeaddedstruture,forinstane,
aresultdue toLazard[14, 1.6℄ saysthatthe ontrationmap S peG(S)!Spe(R ) must
be one-to-one. Lastly, one has a \handle" onthe elementsof G(S) inthat they have the
nite representation mentioned in Theorem 2.2(i). Although this representation is not
unique, itsvery existene isuseful.
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 whih we build G(S). Or it maybe
that S itself is built by a natural onstrution that begins with R . We shall enounter
examples of both sorts. Our rst examplesare algebrai.
3.1. The epimorphi hull of a ommutative semiprime ring [Storrer℄. To
dene this ring we need another ategorial notion. A monomorphism f A ! B is
alled essential if eah map g:B ! C must be a monomorphism whenever g Æ f is a
monomorphism. Usually one disusses essentiality for extensions of modules but the
denition makes sense inany ategory(f [23℄).
An epimorphi hull E(A) for an objet A is an essential monoepif:A! E(A) with
thepropertythatifh:A!B isanessentialmonoepi,thereisamapg:B !E(A)sothat
gÆh =f. Storrer [23℄ showed that epimorphi hulls are unique up to isomorphism and
studiedthe epimorphihullsofommutativesemiprimerings indetail. Let usreallthat
if R isasemiprime ringand S isanextensionof R , thenS isalledaring of quotientsof
R if given s 6=02 R , there exists a t 2 S so that st 2R ;st6=0. The simplestexample
of aring ofquotientsisthe passagefrom Z toQ, or,indeed,fromany domaintoitseld
of frations. 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 ) whih
is unique up toisomorphism over R [11, 2.3℄. It is a regularring [11, 2.4℄ so one has the
following:
epimor-epimorphi regular ontext determined by R and its omplete ring of quotients Q(R ). It
is (up to isomorphism)the only epimorphiregular extension whih is essential.
Thus we have two importantrings assoiated with a generalommutativesemiprime
ringR , |T(R ),due toOlivier,andE(R ),due toStorrer. Eahhas auniversalproperty.
The tworings are relatedas follows:
3.3. Theorem. [Folklore℄ Let R be aommutative semiprime ring. Let T(R )beits
uni-versal epimorphi regular extension, and E(R ) be its epimorphi hull. There is a
homo-morphism from T(R ) onto E(R ) that xes R . (Indeed any epimorphi regular extension
of R mapsonto E(R ) overR .)
Proof. For one thing, suh a map must exist by the universal property of T(R ). But in
fat,thenierwaytoseethisistouseZorn'slemmatohooseinT(R )anidealI maximal
with respet to having (0) intersetion with R . Although the ideal I is far fromunique,
the ring T(R )=I is a regular essential mono-epi, and hene a opy of E(R ). Note that
there are ertain topologialsituationswhen the hoie of the ideal I is\anonial".
3.4. Remark. IamindebtedtoHansStorrerforpointingouttomeinprivate
orrespon-dene the following fasinating point onerning E(S). Sine 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
quasi-inverse of P n i=1 r i s i
and this is given by
n X i=1 r i s i ! = X M Y i2M (1 s i s i ) Y
j2M= s j 0 X k=2M 0 Y
l2(M[fkg)= s l 1 A r k 1 A
where M runs through allsubsets of f1;2;:::;ng.
3.5. Minimal regular rings (Piere, Burgess). In ordertopresent this example,
we must reall some notions from studies by Piere [17℄, Burgess [1℄, and Burgess and
Stewart [3℄.
SupposethatSisaring. TheharateristisubringofS,denoted(S),isthemaximal
epimorphiextension ofthe anonial imageof Z inS. By B(S)one denotes the algebra
of entral idempotents of S. The minimal subring of S (denoted P(S) in [1℄) is the
subringofS thatisgeneratedby(S)andB(S). WhenP(S)=S, S isalledaminimal
ring. The minimal subring of a regular ring is itself regular. A regular ring is minimal,
exatly when its eld images are primitiveelds. Minimal regular rings were studied by
Piere [17℄ who showed that they form a ategory that is ontravariantly equivalent to
the ategory of boolean spaes over the one-point ompatiation of the set of primes
alled the ategory of \labelledBoolean spaes"(LBS). The ategory of minimalregular
rings has some interesting properties,for example, itis omplete and o-omplete.
To relate these notions to epimorphi regular ontexts we suppose that the ring S is
3.6. Proposition. (i) P(S)=G(S),
(ii) S is the lassial ring of quotients of R and hene the regularity degree of G(S)
with respet to R is one.
Proof. (i)Sine G(S) isthe smallestregularring between R andS, 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
spaes using the Piere sheaf [16℄ they have a ommon base spae X = SpeB(S). For
eahx2X thestalksG(S)
x
andP(S)
x
areelds,andG(S)
x
isasubeldofP(S)
x
. Sine
the stalks of P(S) are primitive elds, the stalks of the two rings oinide, as must the
rings themselves.
(ii)Suppose thatr2R . Bythe regularity ofS,the annihilatorof rinS has the form
eS for some idempotent e of S. Sine B(S)=B(R ), the annihilator of r in R iseR and
it is easy to hek that this makes Q
l
(R ) regular. Furthermore eah non zero-divisor
in R is a non zero-divisor, and hene invertible in S. The universal property of Q
l (R )
impliesthatitmapsintoS overR . Sineitsimageisaregularringbetween R andG(S),
it is G(S). The kernel of the map is learly trivial,establishing the isomorphism. Sine
the elements in Q
l
(R ) have the form ab 1
;a;b 2 R , the regularity degree of G(S) with
respet to R is one.
3.7. Remark. It follows that oneannotbuild aounterexample toProblem2.10using
the rings of Proposition 3.6.
3.8. Rings of ontinuous funtions. One is led naturally to onsider rings of the
form C(X), the algebra of all ontinuous funtions on a (ompletely regular) Hausdor
spae. There are two reasons for this: rst, if the algebrai notions are valid, they may
have interpretations of interest in rings of funtions; and seond, rings of funtions are
suÆientlyompliated thatthey are likely toprovideexamples andounterexamplesfor
algebraiquestions. This wasalready learwhen Storrerstudiedthe epimorphihull[23,
11.6℄
Letusrstreallsomebasinotions,bearinginmindthat[5℄isthe standardreferene
for the topi. If X is a topologialset and f:X ! R is a ontinuous funtion, then the
zero-setof f isfx2X jf(x)=0g. A subsetof X is aG
Æ
if itis aountable intersetion
ofopensets. Apointp2XisalledaP-pointofX ifthezero-setoff isaneighbourhood
of p whenever f 2C(X) and f(p)=0.
AtopologialspaeXisalledompletelyregular[5℄ifitisaHausdorspaeinwhih
anypointandanylosedset disjointfromitan beompletelyseparated byaontinuous
real-valued funtion on X. When onsidering the ring of all ontinuous funtions on a
topologialspaeone an assume,withoutlossof generalitythat X isompletelyregular
[5,Chapter 3℄.
3.9. -algebras. There are several simultaneous strutures on C(X)|that of a
om-mutativealgebraoverR, that of apartiallyordered set, andthat ofa lattie. In[8℄these
essential features were abstrated to the notion of a -algebra whose denition requires
A lattie-ordered ring A is alled Arhimedian if, for eah element a whih is dierent
from 0 the set fna:n 2 Zg has no upper bound in A. An element a 2 A +
is alled a
weak orderunit of A of ifb 2A and a_b=0implyb =0. Lastly, anideal ofA is alled
an l-ideal, if a 2 I;b 2 A, and jbj jaj imply b 2 I. A -algebra is an Arhimedean
lattie-ordered algebraoverR inwhih1 isa weak orderunit. EahC(X) isa -algebra
but the onverse is far fromtrue. If A is a-algebra then M(A) denotes the set of
max-imal l-ideals in A endowed with the hull-kernel topology. For eah M 2 M(A), A=M is
a totallyordered eld ontaining R. The ideal M is alled real or hyper-real aordingly
as A=M oinides with R or properly ontains it. The (possibly empty) subset of real
maximall-ideals isdenoted R (A), and if T
R (A)=(0), then Ais asubdiret produtof
opies of R and one alls A a -algebraof real-valued funtions.
Henriksen and Johnson [8, 2.3℄ showed that any -algebra A has a representation as
analgebraofontinuousfuntionstotheextended realsdenedonthespaeM(A). Eah
element of A is real on a dense open subset of M(A) and funtions are identied when
they agree on the intersetion 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 Cauhy sequenes onverge).
A natural and reurring questionis whether a given -algebra is isomorphito aring
of the form C(X). Henriksen and Johnson [8, 5.2℄ found onditions whih are neessary
andsuÆient|theringmustbea-algebraofreal-valuedfuntions,itmust beuniformly
losed, be losedunder inversion, and R (A) must bez-embedded in M(A).
3.10. P-spaes. AtopologialspaeisalledaP-spaeifeveryG
Æ
isopen,equivalently,
if every zero-setisopen. Compat P-spaes areneessarily niteand ountable P-spaes
are disrete, but there existP-spaeswith noisolatedpoints. These spaesare pertinent
beause X isa P-spae preiselywhen C(X) isits own epimorphi hull.
3.11. Almost P-spaes. A spae X is almost P if every non-empty zero-set has a
non-empty interior, equivalently every nonempty G
Æ
set has non-empty interior. Unlike
P spaes, almost P spaes an be ompat and innite, though they also are disrete
if they are ountable. Watson [25℄ has onstruted in ZFC a ompat almost P spae
with no P-points. There is also an interesting spae due to Levy [12℄ who invented
these spaes [13℄. One again there is an algebraiharaterization: it is preisely when
C(X)=Q
l
(X) that X is almostP.
3.12. The delta topology. [18, 1W℄ There is a anonial way of making a spae X
into a P-spae by (if neessary) strengthening its topology. One takes as a base for the
newtopologyallG
Æ
setsintheoriginalspae. Itisequivalenttotakethesetofzero-setsas
a basefor a new topology. The new spae, denoted X
Æ
, is alledthe P-spae oreetion
of X. If j is the ontinuous identity map X
Æ
! X then one has the following universal
property: if Y is a Tyhono P-spae and f:Y ! X is a ontinuous map, then there is
a ontinuous map k:Y ! X
Æ
so that j Æk = f. In ategorial terms, P-spaes form a
3.13. The epimorphi hull of C(X). The lassial and omplete rings of quotients
of a C(X) were onsidered as instanes of the general algebrai study by Fine, Gillman,
and Lambek in their now lassi [4℄. We will use their notationfor these rings, Q
l (X),
and Q(X) respetively. The main resultreads:
3.14. Theorem. [Fine, Gillman,Lambek, [4,2.6℄℄ The ring Q(X) is the set of all
on-tinuous funtions dened on dense open sets of X modulo the equivalene relation that
identies two funtions that agree onthe intersetion of theirdomains. The ring Q
l (X)
is the algebra of all ontinuous funtions on dense ozero sets of X modulo the same
equivalene 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 funtions asfollows:
3.15. Theorem. [Hager [6℄℄
Suppose that the spetrum 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) beause
the symbol E(X) had an existing and oniting meaning in topology. The epimorphi
hull was shown to be a -algebra for all spaes X. The most natural (and still open)
question was:
3.16. Problem. Let X be a Tyhono spae. Find neessary and suÆient onditions
onX for H(X) tobe isomorphi toa C(Y).
3.17. Subproblem. Find neessary and suÆient onditions on X for the -algebra
H(X) tobe uniformlylosed.
Problem3.16has neverbeen solved. Indeed,one doesnot even knowwhether X must
ontain a P-point if H(X) is a C(Y), although this was reently established positively
for basially disonneted spaes [7℄. The presene of a dense set of P-points, indeed of
isolatedpoints,does not suÆe togivethat H(X) isa C(Y) [7℄.
Here is asummary of the main things one doesknow[19℄:
1. IfH(X) isa C(Y),then ompat subspaes of X are sattered.
2. If X has a ountable dense set of isolated points then H(X) = C(N), where N
denotes aountable disretespae.
3. One has a haraterization of when Q
l
(X) isa C(Y).
4. One has many examples of spaes, almost P and not, for whih H(X) is a C(Y),
aswell asexampleswhere some but not allofthe three rings ofquotientsQ
l
(X),H(X)
and Q(X) are isomorphi torings of ontinuous funtions.
AnimportanttoolforstudyingH(X)isthedenitionofthefollowing\test"subspae:
3.18. Definition. By gX denotethe intersetion of the dense ozerosets of a
realom-pat Tyhono spae X.
(ii) IfX has a dense almost P spae it has a largestone and itis gX.
(iii) The following are equivalent:
(a)H(X) isa -algebraof real-valued funtions,
(b) gX is densein X,
() C(X)has aregular ring of quotients of the form C(Y).
This shows that (f Problem3.16):
(iv) If H(X) is aC(Y) then Y =(gX)
Æ
, (gX with the deltatopology).
3.19. A ontext ofreal-valued funtions. Morereentlytherehasbeen parallel
work undertaken by Henriksen, Raphael, and Woods whih drops the rings of quotients
aspet but keeps the disussion very onrete. The idea is the following: Let X be a
Tyhono spae and observe that C(X) lies in the natural von Neumann regular ring
F(X)of all real-valued funtions dened on X. LetG(X)bethe ontext determined by
C(X)and F(X). Like H(X),the ring G(X) isalways a-algebra but unlike H(X) it is
always a -algebra of real-valued funtions.
When X isanalmostP spae, H(X)and G(X)oinide. Ingeneralthese ringsdier
but H(X) isalways ahomomorphiimage of G(X).
One is no longer dealing with rings of quotients, but still has very natural problems
suh as:
3.20. Problem. Find neessary and suÆient onditions onX so that G(X)bea ring
of ontinuous funtions.
This is known toour forsome non P spaes, but it ours so rarely that one raises
a more general question. Let G u
(X) denote the set of limits of Cauhy sequenes from
G(X).
3.21. Problem. Find neessary and suÆient onditionson X forG u
(X) tobe aring.
[Of ourse there is the parallel problem for H(X)℄. It is easy to see that G u
(X) is
losed under sums, nite sups and infs, and salar multiples. Thus in the ases where it
is a ring, it is a -algebra. It is alsoeasy to see that being a ring is equivalent to being
losed under forming squares. Note that Isbell gave anexample [9, 1.8℄ of analgebra of
real valuedfuntions whose uniform losureis not losed under taking squares.
Lastly returning toOlivier's work, there is the question
3.22. Problem. DesribeT(C(X))whereX isanarbitraryTyhonospae. 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 funtions or a uniformly losed
-algebra?
AfterwordReentjointworkwithW.D.Burgessusingalgebraimethodsseemstogive
Referenes
[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, entralidempotentsand thePiere sheaf, Contemp.
Math, 171, 1994,51-67.
[3℄ W.D. Burgess and P.N. Stewart, The harateristi ring and the \best" way to
adjoin a one, J. AustralianMath. Soiety, 47, 1989, 483-496.
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Department of Mathematis,
Conordia University,
Montreal Canada
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