Behavior of countably generated pure-projective modules

Publicacions Ma emá iques, Vol 36 (1992), 401-406 . BEHAVIOR OF COUNTABLYGENERATED PURE-PROJECTIVE MODULES Abs ac GORO AzumAYA Dedica ed o he memo y o P o esso Pe e Menal We i s p o e ha e e y coun ably p esen ed module is a pu e epimo phic image o a coun ably gene a ed pu e-p ojec i e mod- ule, and by using his we show ha i e e y coun ably gene - a ed pu e-p ojec i e module is pu e-injec i e hen e e y module is pu e-injec i e, while i in any coun ably gene a ed pu e-p ojec i e module e e y coun ably gene a ed pu e-p ojec i e pu e submod- ule is a di ec summand hen e e y module is pu e-p ojec i e . Le R be a ing . A le R-module M is called pu e-p ojec i e i e e y pu e epimo phism on o M spli s . As is well-known, M is pu e-p ojec i e i and only i M is a di ec summand o a di ec sum o ini ely p esen ed le R-modules . On he o he hand, M is called pu e-injec i e i o any le R-module in which M is embedded as a pu e submodule M is always a di ec summand o i . We call R a le pu e-semisimple ing o a ing o le pu e global dimension ze o i i sa is ies he ollowing ob iously equi alen condi ions : (1) e e y le R-module is pu e-p ojec i e, (2) e - e y le R-module is pu e-injec i e, (3) o any le R-module M e e y pu e submodule o M is a di ec summand o M . I has been known ha e e y le pu e-semisimple ing is a le A inian ing and besides le pu e-semisimple ings a e cha ac e ized as hose ings R o which he ollowing equi alen condi ions hold : (4) e e y le R-module is a di ec sum o ini ely gene a ed submodules, (5) e e y le R-module is a di ec sum o indecomposable submodules, (6) e e y indecomposable le R-module is ini ely p esen ed ([2], [3], [8], [10]) . The au ho has hen shown in [1] ha e en i we eplace he e ms e e y and any in (1), (2), (3) and (6) by he e ms e e y coun ably gene a ed and any coun ably gene a ed espec i ely we s ill ha e condi ions each equi alen o he le 40 2  G . AzumAYA pu e-semisimplici y o R . Now in his pape we shall gi e u he cha - ac e iza ions o pu e-semisimple ings in e ms o coun ably gene a ed pu e-p ojec i e modules . In o de o con i m hese cha ac e iza ions we need o combine Simson's heo em in [6] ha i e e y coun ably p e- sen ed le R-module is pu e-p ojec i e hen R is le pu e-semisimple wi h he ollowing c ucial p oposi ion, which e ines he known Wa ield's heo em in [7] ha e e y module is a pu e epimo phic image o a di ec sum o ini ely p esen ed modules and whe e a module is called coun - ably p esen ed i i is isomo phic o he ac o module o a coun ably gene a ed ee module modulo a coun ably gene a ed submodule : P oposi ion 1 . Le M be a coun ably p esen ed le R-module . Then he e exis s a le R-module P which is a coun able di ec sum o ini ely p esen ed le R-modules and has a pu e epimo phism P - M . P oo : Since M is coun ably p esen ed, he e exis a (in ini e-) coun - ably gene a ed ee le R-module F, a coun ably gene a ed submod- ule G o F and an epimo phism W : F ~ M whose ke nel is G . Le ul, u2, u3, . . . be a coun able ee basis o F and l, 2, 3, . . . a coun - able gene a o s o G . Fo each posi i e in ege n, we deno e by F n he ini ely gene a ed ee submodule Rul G Ru2 ® . . . ® Ru n o F and by G n he ini ely gene a ed submodule R l + R 2 + - - - + R n o G . Then 00  Ce Cea ly we ha e F = Y~ F n and G = E G n . Mo eo e , o each n, he e n=1 n=1 is an m such ha G n C F n , . Le l(n) be he leas one among such m's, and le m(n) = max(l(n), n) . Then, i is ob ious ha G n C F,,( n ) o e e y n and F=  Fnn(n) . n=1 00 Conside now coun able (ou e ) di ec sums S = ® F, (n ) and T = n=1 ® G n .  Then T is a submodule o S .  Fo each n, we deno e by qn n=1 he n- h canonical embedding F,,( n ) , S, Le ., qn(x), o x E F,,( n ), is he elemen o S whose n- h en y is x and o he en ies a e all 0 . Clea ly he es ic ion o q n o he submodule G n o ,,,(n) is he n- h canonical embedding G n ~ T . Now he ac o module F,)/G, is ini ely p esen ed o each n . Le P= ®(F,  ,(n)/Gn) be he coun able n=1 di ec sum . Then he na u al epimo phisms F  , (n ) - Fm(n)/Gn oge he de ine an epimo phism 0 : 5 --> P whose ke nel is T . I we nex associa e COUNTABLY GENERATED PURE-PROJECTIVE MODULES  403 each s E S wi h he sum n-1 o s, hen we ha e an epimo phism : S  F . The es ic ion o o G gi es clea ly an epimo phism g : T -> G, and so we ha e ha cp( (T)) = W(g(T)) = cp(G) = 0 . Since T is he ke nel o 7P, his implies ha an epimo phism h : P  M is well-de ined so ha h o 0 =W o . We now show ha h is a pu e epimo phism . Le E be a ini ely p esen ed le R-module . Thus he e exis a ini ely gene a ed ee le R-module L and an epimo phism 7 : L --> E whose ke nel K is ini ely gene a ed . Le ca : E -> M be a homomo phism . Conside he p oduc a o 7 : L ---~ M . Since L is p ojec i e, he e mus exis a homomo phism ,l : L --> F such ha cW o ~3 = ca o 7 . F om his ollows ha W(O(K)) = a(7 (K)) = 0 and so O(K) C G, he ke nel o cp . Since bo h L and K a e ini ely gene a ed, hei homomo phic images ~3(L) and O(K) a e ini ely gene a ed submodules o F and G espec i ely . The e o e we ha e (L) C F,(n) and /3(K) C G n o su icien ly la ge n . We ix such an n, and de ine he homomo phism -y : L - S by y= q n o /3 . Then we ha e o y = o qn o ~3, bu Since o qn is clea ly he inclusion map F,(n) ---+ F i ollows ha o y = Q . On he o he hand, ha /3(K) C G n implies ha y(K) = g n (O(K)) C g n (G n ) C T and he e o e 0(y(K)) C O(T) = 0, Le ., K is con ained in he ke nel o 0 o y : L -~ P . This shows ha a homomo phism 5 : E --> P is well-de ined so ha 6o7 = Ooy . We ha e hen hobo7 = ho0oy = cpo oy = cpoo = ao7 , bu Since n : L -> E is an epimo phism we know ha h o 6 = ca . This is ue o e e y ini ely p esen ed module E and o e e y homomo phism a : E - M, and hus i is p o ed ha h is a pu e epimo phism . s nin F, whe e s n (6F,,( n » is he n- h en y Theo em 2 . R is le pu e-semisimple i and only i e e y coun ably gene a ed pu e-p ojec i e le R-module is pu e-injec i e . P oo .. Clea ly we need only p o e he i pa . Le M be any coun - ably p esen ed le R-module . Then by he p eceding p oposi ion he e exis a coun able di ec sum P o ini ely p esen ed le R-modules and an epimo phism h : P ---> M whose ke nel Q is a pu e submodule o P . Now he di ec sum P(` ), N being he se o all na u al numbe s, o he coun able numbe o copies o P is also a coun able di ec sum o ini ely p esen ed le R-modules and hence is coun ably gene a ed and pu e-p ojec i e . Thus by ou assump ion p(N) is pu e-injec i e, o equi alen ly, P is E-pu e-injec i e . The e o e he pu e submodule Q o P is a di ec summand o P ([9, p . 1100], [1, P op . 3 .5]), which means ha M can be embedded in o P as a di ec summand and he e o e M is pu e-p ojec i e oo . Thus i u ns ou ha e e y coun ably p esen ed 40 4  G . AzumAYA le R-module is pu e-p ojec i e . I ollows om Simson [6, Th . 6 .3] ha R is le pu e-semisimple . As is easily seen, e e y coun able di ec sum o coun ably p esen ed modules is coun ably p esen ed oo and so in pa icula e e y coun able di ec sum o ini ely p esen ed modules is coun ably p esen ed . The e- o e, om he p eceding heo em we can de i e he ollowing, which is howe e ega ded as a dual o he aboye e e ed Simson's heo em : Co olla y 3 . R is le pu e-semisimple i and only i e e y coun ably p esen ed le R-module is pu e-injec á e . Now he ollowing is o e ine he equi alen e o he condi ions (2) and (6) in [1, Th . 4 .5] : Theo em 3 . R is le pu e-semisimple i aud only i o any coun - ably gene a ed pu e-p ojec i e le R-module P e e y coun ably gene a ed pu e-p ojec i e pu e submodule o P is a di ec summand o P . P oo . : We need only p o e he i pa oo .  Le M be a coun ably p esen ed le R-module . This means ha he e exis a coun ably gene - a ed ee le R-module F and an epimo phism cp : F --> M whose ke nel G is coun ably gene a ed . On he o he hand, by P oposi ion 1, he e exis a coun able di ec sum P o ini ely p esen ed le R-modules and an epimo phism h : P --> M whose ke nel Q is pu e in P . We shall show ha Q is coun ably gene a ed oo . Fo , since F is p ojec i e he e is a homomo phism : F --> P such ha h o = (p . Le p be an elemen o P . Then h(p) is in M and so we can ind an x E F such ha W(x) = h(p) whence h( (x)) = h(p), Le ., h(p- (x)) = 0 . Thus weknow ha p- (x) is in Q and he e o e ha P = (F) +Q . On he o he hand, i x is in F hen he equali y h( (x)) = cp(x) implies ha (x) is in Q i and only i x is in G, and he e o e we ha e (F) 1 Q= (G) . Thus we know ha Q/ (G) -- P/ (F) . Bu since P is coun ably gene a ed i s homomo phic image P/ (F) whence Q/ (G) is coun ably gene a ed oo, while since G is coun ably gene a ed i s homomo phic image (G) is also coun ably gene a ed . F om his we can conclude ha Q is coun ably gene a ed . Now, since P is pu e-p ojec i e, he coun ably gene a ed pu e submod- ule Q o P is pu e-p ojec i e acco ding o Kielpi íski-Simson [4, Co . 1 .5] . (This can also be p o ed by using he no ion o Mi ag-Le e modules as ollows : As is well-known, e e y pu e submodule o a pu e-p ojec i e module is a Mi ag-Le ie module, and so Q is a Mi ag-Le ie mod- ule, Le ., he canonical homomo phism ( l A Z ) ~ Q , 11 (A ¡ ® Q) is R  R COUNTABLY GENERATED PURE-PROJECTIVE MODULES  405 a monomo phism o e e y amily {Ai} o igh R-modules . Since Q is coun ably gene a ed, Q mus be pu e-p ojec i e by Raynaud-G uson [5, Pa II, Co . 2 .2 .2] .) By he assump ion o ou heo em, we know ha Q is a di ec summand o P and so M can be embedded in o P as a di ec summand, which implies ha M is pu e-p ojec i e . Thus we ha e shown ha e e y coun ably p esen ed le R-module is pu e- p ojec i e, and he e o e again by Simson's heo em ([6, Th . 6 .3]) R is le pu e-semisimple . In his connec ion, i is o be poin ed ou ha Theo em 2 is a di- ec consequence o Theo em 3 . Fo , le P be a le R-module and Q a coun ably gene a ed pu e-p ojec i e pu e submodule o P . Suppose ha e e y coun ably gene a ed pu e-p ojec i e le R-module is pu e- injec i e . Then Q is a di ec summand o P . Thus, by Theo em 3, R is le pu e-semisimple . Rema k . A . Abe has independen ly ob ained P oposi ion 1 and The- o em 2 oo . Indeed, he p o es P oposi ion 1 by using [6, Co . 2 .5] ha i a module M is a di ec limi o modules M i 's hen he canonical epimo phism ® M i --> M is pu e . i Re e ences 1 .  G . AzumAYA, "Coun able gene a edness e sion o ings o pu e global dimension ze o," Rep esen a ions o algeb a, Camb idge Uni- e si y P ess, 1992 . 2 .  S . U . CHASE, Di ec p oduc o modules, T ans . Ame . Ma h . Soc . 9 7 (1960), 457-473 . 3 .  L . GRUSON AND C . U . JENSEN, Deux applica ions de la no ion de L-dimension, C .R . Acad . Sci . Pa is, Sé . A 282 (1976), 23-24 . 4 .  R . KIELPIIVSKI AND D . SIMSON, On pu e homological dimension, Bull . Acad . Polo . Sci ., Sé . Sci . Ma h . As . Phys . 2 3 (1975), 1-6 . 5 .  M . RAYNAUD AND L . GRUSON, C i é es de pla i ude e de p ojec- i e, In en . Ma h . 13 (1971), 1-89 . 6 .  D . SIMSON, On pu e global dimension o locally ini ely p esen ed G o hendieck ca ego ies, Fundamen a Ma h . 96 (1977), 91-116 . 7 .  R . B . WARFIELD, JR ., Pu i y and algeb aic compac ness o mod- ules, Paci ic J . Ma h . 28 (1969), 699-719 . 8 . W . ZIMMERMANN, Einige Cha ak e isie ungen de Ringe, übe denen eine Un e moduln di ek Summanden sind, Si zungsbe , Baye . Akad . 3 (1972), 77-79 . 406  G . AZUMAYA 9 . W . ZIMMERMANN, Rein injek i e di ek e Summen on Moduln, Comm . Algeb a 5 (1977), 1083-1117 . 10 . B . ZIMMERMANN-HUISGEN, Rings whose igh modules a e di- ec sums o indecomposable modules, P oc . Ame . Ma h . Soc . 77 (1979),191-1 .97 . Depa men o Ma hema ics Indiana Uni e si y Blooming on, IN 47405 U .S .A . Rebu el 7 de Gene de 1992