Hopfian and co-Hopfian objects

Publicacions Ma emá iques, Vol 36 (1992), 293-317 . A bs ac HOPFIAN AND CO-HOPFIAN OBJECTS K . VARADARAJAN 1 The aim o he p esen pape is o s udy Hop ian and Co-Hop ian objec s in ca ego ies like he ca ego y o ings, he module ca e- go ies A-mod and mod-A o any ing A . Using S one's ep esen- a ion heo em any Boolean ing can be ega ded as he ing A o clopen subse s o a compac Hausdo o ally disconnec ed space X . I u ns ou ha he Boolean ing A will be Hop ian ( esp . co-Hop ian) i and only i he space X is co-Hop ian ( esp . Hop- ian) in he ca ego y Top . Fo any compac Hausdo space X le C7,(X)( esp . Cc(X)) deno e he R( esp . C)-algeb a o eal ( esp . complex) alued con inuous unc ions on X . Using Cel and's ep- esen a ion heo em we will p o e ha C7¿(X)(Ce(X)) is Hop ian ( espec i ely co-Hop ian) as an R(C)- algeb a i and only i X is co-Hop ian ( espec i ely Hop ian) as an objec o Top . We also s udy Hop ian and co-Hop ian compac opological mani olds . In oduc ion The no ion o a Hop ian g oup [4] is by now classical . Th oughou he p esen pape he ings A we conside a e associa i e ings wi h an iden i y elemen lA =,1= 0 . Any sub ing B o A is equi ed o sa is y he condi ion ha lB = lA . All he modules conside ed a e uni a y modules . A-mod ( esp . mod-A) will deno e he ca e ogy o le ( esp . igh ) A-modules . In [12] V .A . Hi ema h has in oduced he concep o Hop ici y o a ing A ega ded as a ing and also o any MeA- mod . We will show in he p esen pape ha A is Hop ian in A-mod i and only i i is Hop ian in . mod-A (Theo em 1 .3) . When hese wo equi alen condi ions a e sa is ied we will simply say ha A is Hop ian as a module . The e a e ob ious dual no ions o A being co-Hop ian espec i ely as a ing, as an objec in A-mod and as an objec in mod-A . We ob ain a necessa y and su icien condi ion o A o be co-Hop ian in A-mod (P oposi ion 1 .4) . Unlike he Hop ian case, by means o a speci ic 1 Resea ch done while he au ho was pa ially suppo ed by NSGRC g an AS225 . 29 4  K . VARADARAJAN example we show ha có-Hop ici y is no le - igh symme ic . Also, we will gi e examples o show ha A being Hop ian ( esp . co-Hop ian) as a ing and A being Hop ian as a module ( esp . co-Hop ian in A-mod o mod-A) a e independen o each o he . As an immedia e consequence o ou necessa y and su icien condi ion i will ollow ha a nó necessa ily commu a i e in eg al domain A is co-Hop ian in A-mod as well as mod-A i and only i A is a skew- ield . I is a well-known esul ha any noe he ian MeA-mod is Hop ian in A-mod and ha any a inian MeA-mod is co-Hop ian ([17, page 42]) . A gumen s used in p o ing his esul will show ha any ing A wi h a .c .c . on wo sided ideals is Hop ian as a ing and any ing A wi h d .c .c on sub ings is co-Hop ian as a ing . In pa icula any le noe he ian (hence any le a inian) ing A is Hop ian as a ing . Easy examples can be gi en o show ha e en ields need no be co-Hop ian as ings . Simila o he esul ha any le a inian ing is le noe he ian we ha e he esul ha any ing A which is co-Hop ian in A-mod is au oma ically Hop ian in A-mod, hence also Hop ian in nod-A (P oposi ion 1 .10) . Le n be any in ege > 1 . I is easy o p o e he ollowing implica ions : a) M  , (A) Hop ian ( esp . co-Hop ian) as a ing => A Hop ian ( esp . co-Hop ian) as a ing . b) M  (A) Hop ian ( esp . co-Hop ian) in M,, (A)-mod => A Hop ian ( esp . co-Hop ian) in A-mod . The analogue o Hilbe 's basis heo e a is alid o Hop ici y, namely MEA-mod is Hop ian i and only i M[X] is Hop ian in A[X]-mod, whe e X is an inde e mina e o e A . This and he analogous esul o M[[X]] in A[[X]]-mod a e p o ed in Sec ion 2 o he p esen pape (Theo em 2 .1) . We do no know whe he he analogous esul is alid o M[X, X - 1] in A[X, X -1 ]-mod . Fo any non-ze o MeA-mod, i is easy o see ha M[X] ( esp . M[[X]]) is no co-Hop ian in A[X] ( esp . A[[X]])-mod . In Sec ion 3 we a e mainly conce ned wi h he case when A is commu- a i e . Fo he esul s s a ed in he p esen pa ag aph i will be assumed ha A is a commu a i e ing . Then i is well-known [22], [24] ha e - e y .g . (abb e ia ion o ini ely gene a ed) A-module is Hop ian . I can easily be shown ha M  ,(A) is Hop ian in M  ,(A)-mod o mod-M,,(A) o all in ege s n >_ 1 . Ou necessa y and su cien condi ion o A o be co-Hop ian in A-mod (Theo em 1 .3) implies ha A is co-Hop ian in A-mod e-* A is i s own o al quo ien ing . In his case we will p o e ha M n (A) is co-Hop ian in bo h M  ,(A)-mod and mod-M,(A) . We will also p o e ha An is co-Hop ian in A-mód o all n >_ 1 . The p oo o his will depend on an auxilia y esul asse ing ha an A-homomo phism : A' - A' is no injec i e i and only i de is a ze o di iso in A, wha e e be he commu a i e ing A (le ama 3 .1) . I is also well-known Hop IAN AND Co-Hop ]AN OBJE CTS  295 [2], [25] ha e e y .g A-module is co-Hop ian i and only i e e y p ime ideal o A is maximal . We will explici ly cons uc a commu a i e ing A which is i s own o al quo ien ing admi ing a p ime ideal which is no maximal . In pa icula his ing will sa is y he condi ion ha A' is co-Hop ian in A-mod o cach in ege n >_ 1 and he e a e .g A-modules which a e no co-Hop ian . The ing A ha we cons uc will ha e he ollowing addi ional p ope ies : c) A is no noe he ian d) A does no ha e d .c .c o sub ings In [12] Hi ema h shows ha i he Boolean ing o clopen subse s o a compac Hausdo o ally disconnec ed space X sa is ies he condi ion ha A is Hop ian as a ing hen X is co-Hop ian as a opological space . He says he does no know whe he he con e se o his esul is ue . Ac ually we no only show ha he con e se is ue bu we also show ha A is co-Hop ian as a ing i and only i X is Hop ian as a opological space . This is ca ied ou in Sec ion 4 . Le X deno e a compac Hausdo £ space and C(X) deno e ei he C7z (X) o Cc (X) . We ega d Ciz (X) as an R-algeb a and Ce (X) as a C-algeb a and simply w i e " he algeb a C(X )" . Using Gel and's ep- esen a ion heo em we show ha C(X) as an algeb a is Hop ian ( esp . co-Hop ian) i and only i X is co-Hop ian ( esp . Hop ian) in he ca - ego y Top o opological spaces . (Theo em 5 .3) . Wc do no ha e any cha ac e iza ion o compac Hausdo spaces which a e Hop ian ( esp . co-Hop ian) . Howe e i is an easy consequence o in a iance o domain ha compac opological mani olds wi hou bounda y a e co-Hop ian . Among compac mani olds wi hou bounda y i can casily be shown ha ini e se s a e he only Hop ian objec s . Among compac nani olds wi h a non-emp y bounda y he e a e no Hop ian o co-Hop ian objec s . I M is a compac mani old wi h bounda y (9M hen he pai (M, (9M) is a co-Hop ian objec in he ca ego y Top e o pai s o opological spaces . We conclude ou in oduc ion by poin ing ou ha Hil on ; Roi be g e c ., ha e s udied epimo phisms and monomo pliisms in he ho no opy ca ego y and we e led o in es iga ing Hop ian and co-Hop ian objec s in he homo opy ca ego y [10], [18] . Finally we wish o hank he e e ee o in o ma ion on li e a u e . In ac mos o he ma e ial in Sec ion 7 has been poin ed ou by he e e ee . Acknowledgemen s . Pa o his wo k was done while he au ho was isi ing Cen e de Rece ca Ma ema ica, Bella e a in Spain . The au ho would like o hank P o esso Cas elle o c ea ing a e y con- duci e a mosphe e o esea ch . Also while ca ying ou his esea ch he au ho ecei ed suppo om NSERC g an A8225 . 29 6  K . VARADARAJAN 1 . Hop ian and co-Hop ian ings and modules Th oughou we will o mula e ou esul s in he ca ego y A-mod o le uni al A-modules . The e a e ob ious analogous esul s in he ca ego y mod-A o uni al igh A-modules . We i s ix ou e minology and no- a ion . Fo any aeA, £A (a) = {beA 1 ba = 0} and A(a) = {beA 1 ab = 0} . By a le ( esp . igh ) ze o di iso in A we mean an elemen a :~ 0 in A wi h eA(a) 7~ 0 ( esp . A(a) ~L 0) . An elemen aeA will be called a le ( esp . igh ) uni i he e exis s an elemen ceA wi h ca = 1 ( esp . ac = 1) . We call aeA le ( esp . igh ) egula i Wa) = 0 ( esp . A(a) = 0) . I is i ial o see ha any le ( esp . igh ) ze o di iso is ne e a igh ( esp . le ) uni . Also any le egula elemen a which is a le uni is au oma ically a wo-sided uni . De ini ion 1 .1 . M A-mod is said o be Hop ian ( esp . co-Hop ian) i e e y su jec i e ( esp . injec i e) homomo phism : M -> M is an isomo phism . I is wcll-known ha any noe he ian ( esp . a inian) module is Hop ian ( esp . co-Hop ian) [17, Lemma 4, pago ; 41] . P oposi ion 1 .2 . AeA-mod is Hop ian i and only i no le ze o di- iso in A is a le uni in A . This is heo em 9 in [12] . Equi alen ly i is welll-known and easy o see ha AeA-mod is Hop ian i and only i A is di ec ly ini o (i .e . xy = 1 ==> yx = 1) [11] . Theo em 1 .3 . AeA-mod is Hop ian i and only i Ae mod-A is Hop- ian . P oo .. Di ec ini eness is clea ly le igh symme ic . P oposi ion 1 .4 . AeA-mod is co-Hop ian i and only i e e y le egula elemen aeA is a wo-sided uni . P oo£ Immedia e consequence o he ac ha injec i e homomo - phisms : A -> A in A-mod a e exac ly gi en by (a) _ Aa wi h aeA le egula . Examples 1 .5 . Conside he ing A=[Z/2Z  Z/2Z 0Z(2) whe e Z(2) is he 2-localiza ion o Z, namely 2121 = { `cQ 1 n odd } . 71 The elemen  2  EA is easily checked o be igh egula bu no (0 in e ible in A . Hence A is no co-Hop ian in mod-A . 29 8  K . VARADARAJAN (b) Thc ; o ily íng homomo phism o i ( esp . Q) ís he íden i y map . Hence Z and Q a e Hop ian and co-Hop ian as ings . Whilc Z is Hop ian in Z-mod, i is no co-Hop ian in Z-mod . Q is bo h Hop ian and co-Hop ian in Q-mod (hence also in Z-mod) . (c) Fo any ing A he unique ing homomo phism cp : A[X] ~ A[X] ca ying X o X 2 and sa is ying ep 1 A = Id A is an injec i e ing homomo phism which is nó su jec i e . (He e X is an inde e mi na e o e A) . Thus A[X] is no co-Hop ian as a ing, wha e e be he ing A . A simila a gumen shows ha A[X, X -1 ] and A[[X]] a e no co-Hop ian as ings . (d) Le A = K[(Xj]a,i o e a coni nu a i e ing K . Then A is a co nmu a i e ing hence Hop ian in A- nod . Le O : J , .1 be a su jec i e ; nap which is no 1bijec i e . Since J is in ini e such a map exis s . The unique ing homomo phism : A --> A sa is ying 1 K = IdK and (X a ) = Xo(a) is hen a su jec i e ing homomo phism which is no an isomo phism . Thus A is no Hop ian as a ing . (e) Le K be a ield and L= K((Xa)a,j) he ield o a ional unc- ions in an in ini e- ; numbe o inde e mina es . Any ield is Hop ian as a ing . Thus L is Hop ian as a ing . I O : J -> J is any in jec i e map, he e is a unique homomo phism cp : L , L o ields sa is ying cp(X .) = Xq(a) and ep/K = IdK . I O is no bijec i e, hen cp is an injec i e ing homomo phism o L in L which is no su jec i e . Hence L is no co-Hop ian as a ing . Since L i a ield om ema ks 1 .6(a) and (c) we see ha L is bo h co-Hop ian and Hop ian in L-mod . ( ) Fo any simple ing A any ing homomo phism : A- ; B is au oma ically injec i e . Hence e e y simple ing is Hop ian as a ing . F om example (e) abo e wc, see ha a simple ; ing (e en a iel(1) need no be c :o-Hop ian as a ing . (g) Le K be a ield and V an in ini e dimensional ec o space o e K . Le A = EndKV . The e exis K-linea su jec ions : V --> V which a e no injec i e .  Choose such an .  Since V ~ V ~ 0 spli s in K- nod, 3 a K-linea map h : V --> V wi h o h = IdV . This means is a igh uni in A . Since ke 7~ 0 we can choose a, g :V -> V wi h g q¿ 0 and g(V) C Ke . Then gEA sa is ies o g = 0 . Thus is a igh ze o di iso in A which is no a igh uni in A . F om p oposi ion 1 .4 we see ha A is no Hop ian in iod-A and hence also no in A-mod om heo em 1.3 . In case V has coun able dimension i ollows om exe cise 14 .13, page 164 o [1] ha he e a e only wo non-ze o ideals in A= EndKV . Hence A is Hop ian as a ing (see ; p oposi ion 1 .12) . (h) I A = K[(xj~ E j] wi h K any commu a i e ing and J in ini e, om 1 .8(d) we see ha A is Hop ian in A-mod, bu no Hop ian as a ing . When K is a ield and V a coun ably in ini e dimen sional ec o space hen A =EndK V is Hop ian as a ing bu no Hop ian as an A-module . As al eady sien Z is co-Hop ian as a ing bu no as a Z-module . When K is a ield, A = K((X«)«Ei) he ield o a ional unc ions in an in ini e numbe o inde e mi- na es is an example o a commu a i e ing which is co-Hop ian as a module bu no co-Hop ian as a ing om 1 .8(e) . P oposi ion 1 .9 . HOPFIAN AND CO-HOPFIAN OBJECTS  29 9 (i) I A is a ing sa is ying a . e . e o wo sided ideals hen A is Hop ian as a ing . (ii) I A is a ing sa is ying d .c .c o sub ings hen A is o-Hop ian as a ing . The p oo o his p oposi ion is simila o ha o lemma 4, pago 42 o [17] and hence omi ed . P oposi ion 1 .10 . Leí A be a ing wi h he p ope y ha A is co- Hop ian in A-mod . Then A is au oma ically Hop ian in A-mod . P oo - Le a be a le ze o di iso in A . F om p oposi ion 1 .2 we ha e only o show ha a is no a le uni in A . On he con a y i a is a le uni in A, he e exis s an ele ien ccA wi h ca = 1 . Then clea ly c is le egula . Since A is co-Hop ian in A-mod, om p oposi ion 1 .4 we see ha c is a wo sided uni in A . Then ca = 1 implies ha a is he in e se o c and hence a is also a wo sided uni . This con adic s he ac ha a is a le ze o di iso . Rema ks 1 .11 . Hi ema h [12] has al eady obse ed ha a di ec summand o any Hop ian module is Hop ian . The same obse a ion is alid o co-Hop ian modules as well . He ema ks ha he does no know o any example o a Hop ian module wi h a submodule no Hop ian . La e in Sec ion 3 we will cons uc such modules . Q is Hop ian and co- Hop ian in Z-mod, he quo ien s Z _ o Q a e no Hop ian in Z-mod . La e esul s in Sec ion 3wil also show ha quo ien s o co-Hop ian modules need no be co-Hop ian' . P oposi ion 1 .12 . Leí A be a ing and n an in ege > 1 . Then (i) M  (A) Hop ian ( esp . co-Hop ian) as a ing ==> A Hop ian ( esp . co-Hop ian) as a ing . (ii) M  ,(A) Hop ian ( esp . co-Hop ian) in M,(A)-mod => A Hop ian ( esp . co-Hop ian) in A-mod . 30 0  K . VARADARAJAN (iii) M,,(A) Hop ian ( esp . co-Hop ian) in A-mod => A Hop ian ( esp . co-Hop ian) in A-mod . P oo . (i) is an immedia e consequence o he obse a ions ha i : A , A is a ing homomo phism, M ( ) : M n (A) --> M,, (A) de ined in he ob ious way is a ing homomo phism and ha M, ,( ) is su j jec i e ( esp . injec i e) ~ is su jec i e ( esp . injec i e) . (ii) simila o (i) abo e excep o he obse a ion ha i : A -> A is a map in A-mod, hen ,,( ) : M n (A) ~ M n , (A) is a map in M n (A)-mod . (iii) is immedia e om he ac ha A is a di ec summand o M n (A) in A-mod . The con e ses o (ii), (iii) a e no ue in gene al . Coun e examples will be gi en in Sec ion 7 . Bu when A is commu a i e o (ii), (iii) he con e ses a e ue and hey will be p o ed in Sec on 3 . We do no know whe he he con e se o (i) is ue . Gi en any non-ze o M6A-mod i is known ha any in ini o di ec sum o copies o M is nei he Hop ian no co-Hop ian in A-mod . Any such module will admi he module N = ® >,M,, as di ec summand whe e M,, = M o each n >_ 1 . The shi map s + which ca ies he n h copy o M o he (n + 1)s copy iden ically is an injec i e map which is no su jec i e . The shi nap s_ which maps he (n + 1)" copy o he n h copy iden ically o n _> 1 and which maps he l` copy o M o ze o is a su jec i e map which is no injec i e . This ac will be mide use o by us la e in Sec ion 3 o cons uc ing a Hop ian module admi ing a non- Hop ian submodule . An in ini o di ec sum o no -ze o modules could e y well be simul aneously Hop ian and co-Hop ian . I P deno es he se o all p imes, M =® PE p(i/pi) is easily seen o be simul aneously Hop ian and co-Hop ian in i-mod, P oposi ion 1.13 . Le A[G] deno e he g oup ing o a g oup G o e he ing A . I A[G] is Hop ian ( esp . co-Hop ian) as a ing hen A is Hop ian ( esp . co-Hop ian) as a ing and G is Hop ian ( esp . co-Hop ian) as a g oup . P oo . Le : A - A be a homomo pl ism o ings and cp : G -+ G a homomo pl ism o g oups . Then he nap ,i : A[G] -> A[G] de ined by 0 (I : g ,c asg) = I :g,c (a y)W(g) 1 is a ing homomo phism . Also i is easily checked ha 9 is su jec i e ( esp . injec i e) <=> and cW a e su jec i e ( esp . injec i e) . P oposi ion 1 .13 is an easy consequence o hese ac s . HOP IAN AND CO-HOPFIAN OBJECTS  30 1 Rema ks 1 .14 . I : A -> A is a map in A-mod and W :G -> G is a g oup homomo phism i is in gene al no ue ha /d :A[G] --> A[G] de ined ~ (I :,EG a,g) = z9Ec (as)W(g) will be a map in A[G]-mod . In case cp = MC i is ue ha Q is a map in A[G]-mod . The analogue o 1 .13 o module ca ego ies is no alid . I A is any commu a i e ing and G any abelian g oup, A[G] is Hop ian in A[G]-mod . G need no be Hop ian . Howe e , he ollowing can be p o ed . P oposi ion 1 .15 . I A[G] is Hop ian ( esp . co-Hop ian) in A[G]- mod, hen A is Hop ian ( esp . co-Hop ian) in A-mod . P oposi ion 1 .16 . Le {Aa}a,J be any amily q ings and A = IIa,jAa hei di ec , p oduc . (i) A is Hop ian ( esp . co-Hop ian) in A-mod <~ :> each A a is Hop ian ( esp . co-Hop ian) in A,-mod . (ii) I A is Hop ian ( esp .  co-Hop ian) as a ing hen each A u is Hop ian ( esp . co-Hop ian) as a ing . P oo .: (i) is an immedia e consequence o he ac ha any map : A - A in A-mod is uniquely o he o m II a : IIA a -> IIA a wi h a : A a -> A a a map in A a -mod and is su jec i e ( esp . injec i e) ~- :> each a is su jec i e ( esp . injec i e) . (ii) I a : A<, -> A,, is a ing homomo phism o each aE .I, hen = II a : IIA a -> IIA, is a ing homomo phism . Mo eo e is su jec i e ( esp . injec i e) i and only i each a is su jec i e ( esp . injec i e) . (ii) is an immedia e consequence o hese ac s . Ac ually p oposi ion 1 .16(1) can be imp o ed as ollows : P oposi ion 1 .17 . Le {Aa}a,J be any amily o ings and A = II .EJAa hei di ec p oduc . Le M a cA .-mod o each acJ . I M = II á ,jMa wi h A-ac ion de ined by a .m = (aama)c .l u)hene e a= (ac )aEJ wi h a a E A a and m = (ma)aE,1 wi h mc, E M a . Then M is Hop ian ( esp . co-Hop ian) in A-mod i and only i each M a is Hop ian ( esp . co-Hop ian) in A,,-mod . Again his is an immedia e éonsequence o he ac ha any map : M - M in A-mod is uniquely o he o m II a : IIM a - IIM a wi h ,, : M a - M aa map in A a -mod . 2 . Hop ici y o he modules M[X],M[X]/(Xn) and M[[X]] Gi en any MEA-mod and an inde e mina e X o e A we de ine 30 2  K . VARADARAJAN M[X]eA[X]- nod, M[X]/(X7')EA[X]/(X7')- nod a id M[[X]]c-A[[X]]- nod as in Sec ion 1 o [23], whe e A[X] deno es he polynomial ing, A[X]/(Xn) o any in ege n >_ 1 deno es he unca ed polynomial ing and A[[X]] he o mal powe se ies ing . The main esul p o ed in his sec ion is he ollowing : Theo em 2 .1 . Le , McA-mod . Then he olowing a e equi alen . (1) M is Hop ian in A-mod . (2) M[X] is Hop ian in A[X]-mod . (3) M[X]/(X n ) is Hop ian in A[X]/(X")-mod . (4) M[[X]] is Hop ian in A[[X]]-mod . P oo .. (2) => (1) . Le : M -+ M be any su jec i e map in A- mod .  Then [X] : M[X] -~ M[X] de ined by [X] (Ek j=o ajXj) _ a  Xj is a su , ec i e ma  in AX mod . Since M[X] is Ho lan in A[X]-modwe see ha [X] is injec i e . This i imedia ely yields he injec i i y o . The p oo s o (3) ==> (1) and (4) => (1) a e simila and omi ed . (1)  (2) . Le W : M[X] - M[X] be any su jec i e A[X]-honio pl ism . Le 0 = coI M :M - M[X] . Then 0 is an A-honio no phism : Mo eo e k  k (4)  cp ajXj= X'O(aj) . j=0 j=o Fo any i > 0 le pi : M[X] -~ M be de ine(¡ by i i<k i > k . Then p i : M[X] - M is a map in A- nod o each i > 0 . Since ep is su jec i e, gi en any ceM he e exis an elemen E~-o a j Xy E M[X] Wi h W(L_j=o a j Xj ) = c . Using 4, we see ha he "cons an e co" o O(ao) is c o equi alen ly pood(ao) = c . This shows ha he mala po~oO :M , M is a su jec i e nap in A- nod . The Hop ian na u e o M in A- nod iniplies ha poOOM -> M is an iso no phism, in pa icula injec i e . Ou ai i is o show ha cp :M[X] - M[X] is injec i e . Le z :k j=o b j XjE M[X] sa is y W(j :k =o bjXj) = 0 . Using 4 and obse ing ha O(bj) _ HOP IAN ANn Co-HOP IAN o13JECTS  309 se I,, = {C e B(X) I x 1C} hen x 1-- x is a hoeomo phism o X wi h max Spec B(X) . De ini ion 4 .3 . A opological space X is said o be Hop ian ( esp . co-Hop ian) in he ca ego y Top i e e y su jec i e ( esp . injec i e) con- inuous map : X - X is a homeomo phism . The main esul o his sec ion is he ollowing . Theo em 4 .4 . A Boolean ing A is Hop ian ( esp . co-Hop ian) as a ing i and onl y i XA is co-Hop ian ( esp . Hop ian) in he ca ego y Top . P oo .. Immedia e consequence o p oposi ion 4 .2 . Rema ks 4 .5 . Le J be any ini e se . The p oduc space H" whe e H = {0, 1} is nei he Hop ian no co-Hop ian . As a se H j is he se o all maps o .7 in o H . Fo any se heo e ic map 0 : J - J we ha e an induced nap ~--> o 0 o H j in o Hj, which is easily seen o be con inuous . I 0 is an injec i e ( esp . su jec i e) map which is no a bijec ion, hen ~--> o 0 s a su jec i e ( esp . injec i e) map which is no bijec i e . Since A = B(H j ) is a commu a i e ing in which e e y p ime ideal is maximal, all - gA-modules a e simul aneously Hop ian and co-Hop ian in he ca ego y o A-modules bu A is nei he Hop ian no co-Hop ian as a ing . I would be nice o cha ac e ize comple ely he Hop ian ( esp . co- Hop ian) compac Hausdo o ally disconnec ed spaces . 5 . Hop ian and co-Hop ian unc ionalgeb as Le K be a commu a i e ing and K-alg deno e he ca ego y o K- algeb as . De ini ion 5 .1 . A K-algeb a A is said o be Hop ian ( esp . co- Hop ian) as a K-algeb a i any su jec i e ( esp . injec i e) K-algeb a homomo phism :A , A is isomo phism . Le R ( esp . C) deno e he ield o eal ( esp . complex) numbe s wi h he usual opology . Fo any compac Hausdo space X le CR (X) ( esp . Cc(X)) deno e he R ( esp . C)-algeb a o con inuous unc ions ' om X o R ( esp . C) . Using he Gel and esp esen a ion heo em we will de e mine necessa y and su ñcien condi ions o Cn(X) ( esp . Cc(X)) o be Hop ian o co-Hop ian in he ca ego y R-alg ( esp . C-alg) . We will mainly concen a e on CR(X) . Simila esul s a e alid o Cc(X) . X deno es a compac Hausdo space and C(X) deno es he R-algeb a CR (X) . I is well-known ha he map x -, ( (x)) EC(x) is a opological 31 0  K . VARADARAJAN imbedding o X in o II_  EC(x)R wi h he ca esian p oduc opology, whe e R = R o each E C(X ) . Also x ~-- m~, = { E C(X) l (x) = 0} is a bijec ion om X o he se o maximal deals in he R-algeb a C(X) . I X~ Y is a con inuous mal) o compac Hausdo spaces, he e is an induced homomo phism cp * : C(Y) - C(X) in R-alg gi en by p* (g) = goce o e e y g E C(Y) . Also gi en any R-algeb a homomo - phism a : C(Y) -+ C(X), he e is a unique con inuous mapW :X - Y sa is ying a = cp* . To see his, o any x E X, a-' ( ¿, ;) is a iaximal ideal o C(Y) and hence a - (M X ) = m,,(=) o a unique elemc w(x) E Y . I jx :X - ; II EC(x)R and jy : Y -> IIg E C(Y)R,, deno e hc : imbeddings jx(x) _ ( (x)) EC(x) and jy(y) = (g(? ))IEC(y)y espec i ely, hen he se heo e ic map cp : X -> Y ob ained abo e sa is ies he condi ion ha diag am 5.2 Y II  a T7 ~ 1- ~ IE I I n  . Jc(y) i~,g is comniu a i e, whe e a*(( ) eC(x)) = (S9)9EC(Y) wi h sg = a(g) . Since a* composed wi h any p ojec ion IIR g --> R g s con inuous we see ha a* is con inuous, hence cp is con inuous, p o ided we check he comniu a i i y o he diag am 5 .2 . Bu i is s aigh o wa d . Thus he se o R-algeb a ho no io phisnis C(Y) - C(X) is he sa c as he se {W* :lW :X -> Y con inuous } . The esul s quo ed so a a e well-known ([20, pagos 327-330] ) . P oposi ion 5 .2 . Le W:X -> Y be a con inuous map o compac Hausdo ' spaces . Then (i) cp* :C(Y) --> C(X) is injec i e <=> W:X --> Y is su jec i e . (ii) cp* :C(Y) -> C(X) is su jec i e ~-¿ W:X --> Y is injec i e . P oo (i) Suppose cp :X -+ Y is no su jec i e . ThenW(X) is á p ope closed subse o Y . We can pick an elemen b E Y - W(X) . Le h :cp(X) -> R be any con inuous unc ion . Then we can ge con inuous ex ensions g :Y -> 1Z, g2 :Y --> R o h wi h gi(b) = 0 and g2(b) = 1 (by Tie ze ex ensio i heo em) . Then gl :~ 92 in C(Y) bu cP*(gi) = h o co = cP*(g2) Since g l ~o(X) = g21 ;G(X) = h . Thus ep no su jec i e = :> cp* no injec i e o equi alen ly W* injec i e => co su jec i e . HOP IAN AND CO-HOPC'IAN OT3 .II CTS  311 I cp :X - Y is su jec i e, hen o any wo se heo e ic maps g :Y -> R, 92 : Y ~ R, we ha e hc ; implica ion g o ;o = 92 o cp ==> g = .g2 . In pa icula his implica ion is ue wi h g , g2 in C(Y) . This p o es (i) . (ii) Suppose cp is no i jec i e, say x jA x2 in X sa is y W(x1) = cp(x2) . Any E C(X) o he o m g o cp wi h g E C(Y) has o sa is y (X1) = (x2) . Howe e , we do know ha 3 E C(X) wi h (xi) = 0 and (X2) = 1 . Thus ep* :C(Y) -> C(X) is no su jec i e . Con e sely, assume ha cp is i jec i e . Then cp :X - W(X) is a ho e- omo phism in(¡ W(X) is closed in Y . Gi en any E C(X), h :W(X) -> R de ined by hW(x) = (x) is con inuous . By Tic ze ex ension lico c n, he e exis s ,g E C(Y) wi h ,gl~o(X) = h . Then cp*(g) = , sl owing ha cp* :C(Y) - C(X) is su j jec i e . Theo em 5 .3 . Le , X he a, compac Hausdo ; ' space . Then C(X) is Hop ian ( esp . co-Hop ian) as an R-algeó a i and on1y i X is co-Hop ian ( esp . Hop ian) as a opologicall space . P oo : Immedia e consequence o p oposi ion 5 .2 . 6 . Hop ianand co-Hop ian objec s in Top among compac mani o1ds Fo each in ege n >_ 1 le D" deno e an n-disk . We may ake D' = {x E R"1 11 . , 11< 1} whe e 11 x 11 deno es he usual no m in R" . B,y de ini ion D ° consis s o a poin . Fo n > l, hc " nap . , ~-- Zx is a con inuous i ,jc ;<aion whic ;h is no a su jec ion . The nap H :D" - " D" gi en by 2x  'o jj x jj <  2 0(x) = ~~  o ~jx jj_ XI 1  2 is a con inuous su jec ion which is no i jec i e . Thus D" is nei he Hop ian no co-Hop ian o n >_ 1 . Obse e ha B :D" , D" clc ined abo e has he acldi ional p ope y ha BIS` = Id,_, . Le M" be ; any compac opological mani old (wi h o wi hou bounda .y) o di ension n > 1 . Then i bedding a disk D" in M" wc can de ine a con inuous su jec ion :M" . - " M" wi h i(M"_ In D") = Id(m__i ,cl)_) and . ¡D" a con inuous su jec ion o D" wi h i sel sa is 'ying . ¡S" - = Id,_, and ID" no i jec i e . I ollows ha M" is no Hop ian . Thus wc- ; ob ain he ollowing . 31 2  K . VARADARAJAN P oposi ion 6 .1 . Th,e only compac , mani olds (wi h o wi hou ,) bounda y which a e Hop ian a e ini e disc e o spaces . As usual o any opological space X we deno e he se , o a cwise connec ed componen e o X by IIO(X) . Theo em 6 .2 . Le Mn and Nn be a compac opological mani olds o he sane dimension n >_ 1, boíh o hem wi hou bounda y . Suppose u he jIIo(Mn)J = III°(Nn)j . Then any con inuous injec ion :M ~N is a homeomo phism . P oo . .- Since Mn and Nn a e compac wc : see ha III0(Mn)J=IIIo(Nn)1 < oo . Lc : {Mi-} k deno e he se o connec ed componen e o M n . Each Mi is a compac connec ed mani old wi hou bounda y, o dimension n . Hence (Mi`) is a compac , connec ed subse o N n . Since ; is injec i e we see ha ¡M` :Mi` -+ (Mi`) is a ho eo no phism . By i i a iance o do nain i ollows ha (M`) is open in Nn . Tlms (M`) is open and closed in N" and aleo connec ed . He ice (Mi`) is a connec ed co iponen o Nn . F o i he injec i i y o i ollows ha i i :,I : j, (Mi') and (M j `) a e dis inc connec ed componen e o N"` . Since ; IIIo(N"`)1 = 1IIo(M")1 = k < oo, i ollows ha { (Mi)}~ a e all he connec ed componen s o Nn, lienee :Mn -> N' is in o . F om he compac Hausdo na u e o M and N wc ; see ha : M" - N" is a homeo mo phism . As an in media e consequence o heo em 6 .2 we ge Co olla y 6 .3 . Any compac mani old M" wi laou bounda y is co- Hop ian in Top . P oposi ion 6 .4 . Any compac mani old M' wi h a, non-emp y bounda y 8M is ne e ' co-Hop ian in Top . P oo . .- By Mo on B own's colla ing l eo em, he e exis s a homeo- mo pl s 0 :BM x [0, 1] -> W whe e W is a neighbou hood o áM in M, sa is ying B(x, 0) = x . o all :L E &M . Le :M --> M be de ined by ( ,) = u o all u E M - 0 M x [0, 1)), (0 (x, )) = 0(x,, ") o all xE W and , E [0, 1] . Then :M , M is a con inuous i jec ion which is no a su jec ion . Le Tope deno e he ca ego y o pai s o opological spaces . De ini ion 6 .5 . A pai (X, A) E Top e is called Hop ian ( esp . co- Hop ian) i any su jec i e ( esp .  injec i e) map :(X, A) - (X, A) o pai s is a ho neomo pl ism . Fo any spaee X le H j (X) deno e hc : singula ho nology wi h in ege coe ñcien s . HoPPIAN AND Co-Hop IAN on,I ;e s  313 Theo em 6 .6 . Le , M, N be compac , mani olds wi h, bounda y sa is- ying he ollowing condi ions . (i) dim M = di n N (ii) ank Ho(OM) = ank Ho(OM), ank Ho(M) = ank Ho(N) and ank H o (M, (9M) = ank HO(N, &N) . Then any injec i e con inuous map :(M,am) - (N,aN) is a homeo- mo phism . P oo .. Le V deno e he double M + Uam M_ o M . Le = ank Ho(M) and s = ank Ho(M, (9M) . I i :&M ,M deno es he inclusion, hen om he , exac sequcnce Ho(¿)M) i Ho(M) - Ho(M, 9M) - 0 we see ha ank Im i = - s . F om he Maye -Vie o is sequence Ho(aM) lx+  1 Ho(M+) ® Ho(M-) - HO(V) -' 0whe e i +:c7M M + , i_ :áM , M_ a e he espec i e inclusions, we sce l a ank HO(V) = 2 - ank o i nage ((i + )   Howe e I n((i+)  (i_),) ¡s - he same as he diagonal subg oup o Im i * ® Im i  he ce has he same ank as Im i, . Thus ank Ho(V) = 2 - ( - s) = + s . Si nila ly i W deno es he doulbe N + Uai N_ we ha e ank H o (W) = + s . In pa icula we ge 1 o«)M)1 = ank Ho(9M) = ank H o (8N) _ 1 a( 7N)1 and 1 o(V)1 = + s= 1 o(W)1 . lOM :OM , iN is a .n injec i e con inuous nap and 1 o(l9M)1 = l o(aN)1 . Henee heo e 6 .2 iníplies ha JOOM is a ho neo no phism . The e is a well-de ined con inuous map g :V -> W sa is ying gIM + :M + -> N + and glM- :M- - N- a e he same as . Then g is injec i e and l7 o(V)j = j7 o(W)j . F om l eo em 6 .2 again we see ha g : V - W is a homeomo phism . I ollows i media ely ha : (M, (9M) -> (N,BN) is a homeomo phism . Co olla y 6 .7 . I M is any compac , mani old wi h bounda 'y OM hen (M, ¿9M) is a co-Hop a e ohjec in Top e . P oo . I n nedia e consequence o heo em 6 .6 . Theo em 6 .8 . (i) I M is any compac mani old wi hou bounda y hen C(M) is Hop ian in R-alg . (ii) I M is an,y compac , opological mani old wi h a, non-em ) y bou ad- a y OOM, hen C(M) is nei he - Hop can no co-Hop ian in R-alg . (iii) I M is a compac , mani old, hen C(M) is co-Hop ia in R-alg i and only i M is a, ini e se , . 31 4  K . VARADARAJAN P oo . : I nmedia e consequence o heo em 5 .3, co olla y 6 .3 and p opo- si ions 6 .1 ancl 6 .4 . 7 . Some ela ed esul s and coun e examples Recall ha a ing A is said o be le ( esp . igh ) 7 - egula i gi en any aeA he e exis s a,n elemen bEA and an in ege n > 1 sa is ying a'L = ba"' ( esp . a' = a"+ l b) . F . Dischinge [7],[8] has sl own ha 7 - egula i y is le igh syn me ic . Be o e Dischinge ob ained his esul , G . Azumaya [3] e e ed o a ing which is bo h le and igh 7 - egula as a s ongly 7 - egula ing . By Dischinge 's esul A is le 7 - egula i and o ily i A is igh n- egula i and o ily i A is s ongly 7- egula . In [7], [8] Dischinge also ob ained he ollowing esul s : 1 . A ing A is s ongly i - egula i and only i e e y cyche le ; o igh A-module is co-Hop ian . 2 . Fo a ing A lhe 'ollowing condi ions a e equi alen . (i) E e y ini ely gene a ed le A-module is co-Hop ian . (ii) E e y ini ely gene a ed igh A-module is co-Hop ian . (iii) M,,(A) is s ongly 7 - egula 'o all in ege s n > 1 . In [9] K .R . Goodea l in oduced he concep o a le epe i i e ing . A ing A is said o be le epe i i e i gi en any aeA and any g le ideal I o A, he le ideal Ia" is . - g . One o hc ; esul s p o ed by Goodea l in [9] is he ollowing : 3 . E e y -gMcA- nod is Hop ian i and only i M,, (A) is le epe - i i e o all in ege s n > 1 . A good epo on hesc : ques ions including new p oo ;s and ew esul s can be 'omid in [13] . Examples 7 .1 . (a) I is clea ha MeA-mod Hop ian => En(¡ (AM) di ec ly ini e . In [19] J .C . S ephe dson gi es examples o di ec ly ini e A wi ii M, .,, (A) no di ec ly ini e o Bo ne in ege n >_ 2 .  Fo any such ing A, wc ; lla e , A Hop ian in A- nod . Also M,,(A) is no Hop ian in M,,(A)- nod . Since A" is a di ec su imand o Mk(A) in A- nod, whene e k 2 > n, we also sc :e ha Mk(A) is no hop ian in A-mod whene e k 2 > n . (b) In pa C o [5] G .M . Be gman cons uc s o each in ege n > 1 a ing A wi h he p ope y ha all egula elen en s in A a e in e - ible, bu M"(A) is no i s own classical ing o quo ien s . In [13] P . Menal cons uc s a ing A which is i s own classical quo ien ing bu M,,(A) is no O e, henee M,,(A) does no ca en ha e a classical ing o quo ien s . A ca e ul inspec ion shows ha in bo h HOP IAN ANn Co-Hop IAN Oi3JECTS  315 hese examples he le egula and he igh egula elemen s o A coincide . Hence by p oposi ion 1 .4 in ou p ese i pape A is co- Hop ian in bo h A-mod and mod-A . Howe e , M,,(A) is nei he co-Hop ian in M,,(A)-mod no co-Hop ian in mod-M,(A) . (c) In sec ion 5 o [16] i is ema ked ha W .L . May has a me hod o ob aining an in ini e abelian Hop ian g oup G such ha he complex g oup algeb á C(G) is no Hop ian as a C-algeb a, hence no Hop ian as a ing . In his example C is Hop ian as a ing, G is Hop ian as a g oup bu C[G] is no Hop ian as a ing . 8 . Open p oblems 1 . I A is Hop ian as a ing, is A[X] Hop ian as a ing? 2 . I A is c :o-Hop ian as a ing and G a . co-Hop ian g oup is A[G] c :o-Hop ian as a . ing? 3 . I A is Hop ian in A- nod and G a Hop ian g ouli is A[G] Hop ian in A[G]- nod? 4 . I A is co-Hop ian in A-mod and G a co-Hop ian g oup is A[G] co-Hop ian in A[G]-mod? 5 . I A is Hop ian ( esp . co-Hop ian) as a ing is i ue ha M,, (A) is Hop ian ( esp . co-Hop ian) as a ing? 6 . Cha ac e i e he Hop ian ( esp . co-Hop ian) objec s in Top a iong compac Hausdo o ally disconnec ed spaces . 7 . I M E A- nod is Hop ian is M[X, X -i ] Hop ian in A[X, X - i]- nod? Re e ences 1 . F.W . ANDERSON ANn K .R . FULLE , "Rings and . Ca egoHes o Modules," G adua e Tex s in Ma he na ics 13, Sp inge -Ve lag, New Yo k, 1973 . 2 .  E .P . A imENDARIZ, Jo ;,, W . FisilER AND Rom ;i L . SNini iz, On ¡Ajec i e and su jec i e cndomo plüs is o ini ely gene a ed mod- ules, Communica ions in A .lgeb a 6 (1978), 659-672 . 3 .  G . AZUMAYA, S ongly n- egula ings, J .Fac . Se¡ ., Hokkaido Uni . 13 (1954), 34-39 . 4 .  G . BAUMSLAG, "Topics in Abelian G oups," Edi ed by J . I win and E .A . Walke , Sco Fo esmann and Company, 1963, pp . 331-335 . 5 . G .M . BE1 CMAN, So ne examples in PI Ring Theo y, Is ael J Ma h . 18 (1974), 257--277 . 31 6  K . VARADARAJAN 6 .  A .W . CI ATTCRS AND C .R . HAJARNAVIS, "Rings wi h chain con- di ions," Resca ch No es in Ma hc na ics 44, Pi nian Publishing Limi ed, 1980 . 7 .  F . DIscilINGCR, Su les anneaux o e emen n- egulie s, C .R . Acad . Sci . Pa is, Se . A 283 (1976), 571-573 . 8 .  F . DISDIIINGCR, S a k 7 - egula Ringe, Disse a ion, Ludwig-Ma- ximilians-Uni e si a , Munchen, 1977 . 9 .  K .R . GOODEARL, Su jec i e endomo phisms o ini ely gene a ed modules, Comm . i n Algeb a 15 (1987), 589-609 . 10 . P .J . MLTON AND J . ROITBGRG, Rela i e epimo phisms and mono- mo phisms in homo opy heo ey, Composi ion Ma hema ica 61 (1987),353-367 . 11 . Y . HIRANO, On Fil ing's Lem na, Hi oshima Ma h . J . 9 (1979), 623-626 . 12 . V .A . HIREMATIi, Hop ian Rings and Hop ian Modules, Indian J . Pu e and Appl . Ma h . 17 (1986), 895-900 . 13 .  P . MENAL, Cancella ion modules o e egula ings, P occedings in ing Tl eo y, G anada,, 1986 SLNM 1328 . 14 .  P . M NAL, Mo i a equi alen e and quo ien ings, Resul s in Ma h . 13 (1988), 137-139 . 15 .  M . ORZGGII, On o endo no pliisms a e iso no pliisms, Ame . Ma h . Mon hly 78 (1971), 357-362 . 16 .  M . OI z cII AND L . RIBCS, Residual ini eness and he Hop p op- e y in Rings, J . Alg . 15 (1970), 81-88 . 17 . P . RIBENBOIM, "Rings and Modules," T ac s in Ma hema ics 24, In e science Publishe s, 1969 . 18 . J . ROI' BERG, Residually ini e Hop ian and co-Hop ian spaces, Con- e np . Ma h . 37 (1985), 131-144 . 19 . J .C . ST PII RDSON, In e ses and ze o di iso s in ma ix ings, P oc . London Ma h . Soc . (1951), 71-85 . 20 . G .F . SIMMONS, "In oduc ion o Topology a,nd Mode n Analysis," McG aw-Hill Book Company, 1963 . 21 .  M .H . STONE, Applica ions o he heo y o Boolean . Rings o Gen- e al Topology, T ans . A .M .S . 4 1 (1937), 375-481 . 22 . J .R . STROOK R, Li ing p ojec i es, Nagoya Ma h . J . 27 (1966), 747-751 . 23 . K . VARADARAJAN, A gene aliza ion o Hilbe 's Basis Theo em, Communica ions in Algeb a 10 (1982), 2191-2204 . HOi'FlAN AND CO-HOPFIAN O 3JGCTS  317 24 . W . VASCONCGLOS, On ini ely gene a ed la modules, D-ans . A .M .S . 138 (1969), 505-512 . 25 . W . VASCONCCLOS, Injec i e endomo phisms o ini ely gene a ed modules, P oc . A .M .S . 25 (1970), 900-901 . The lJni e si y o Calga y Calga y, Albe a LANADA T2N 1N4 Rebu el, 6 de Feb e de 1992