Rings with chain conditions on projective ideals

Pub . Ma . UAB Vol . 26 N° 2 Juny 1982 RINGS WITH CHAIN CONDITIONS ON PROJECTIVE IDEALS 1 . In oduc ion by JOHN KOEHL Depa men o Ma hema ics Louisiana S a e Uni e si y Rebu el 2 d'oc ub e del 1981 A ing R is said o sa is y n igh a .c .c(6 .g .) p i R sa i~ ies he ascending chain condi ion on ( ini ely gene a ed)p ojec i e igh ideals ; n igh d .c.c .(J .g .)P is de ined simila ily . In sec ion wo we show ha i R sa is ies igh d .c .c .P and i U is a wo-sided idealwhich is also a mini- mal p ojec i e igh ideal, hen ei he U 2 = 0 o U 2 = U(2 .3) . I R is commu a i e and U is ini ely gene a ed, hen U 2 = U and is gene a ed by an idempo en elemen (2 .5) . We alsoshow ha i R sa is ies igh a .c .c .P o igh d .c .c .P hen e e y p ojec i e igh ideal is coun ably gene - a ed  (2 .9) . The polynomial (also powe se ies) ing in in- ini ely many a iables o e a ield a e examples o non-Noe- he ian ings sa is ying igh a .c .c .P . The symbols EP a e used o deno e enough pnojec i- ee,me e y nonze o igh ideal con ains a nonze o p ojec i e igh ideal ; and MEP deno es he condi ion ha e e y nonze- o igh ideal con ains a nonze b ini ely gene a ed p ojec i- e igh ideal . 2 . S uc u e o Minimal P ojec i e Ideals I Ris a commu a i e ing sa is yin d .c .c .P hen R has minimal p ojec i e ideals . The nex heo em desc ibes how hese ideals a e ela ed o he o he p ojec i e ideals . We use he ollowing known esul . 2 .1 Lemma . Le- R and S be n .íngs, le . P be a p ojec í eR-module and le Q be an (R,S)- bímodule ha íe p ojec í e ab an  S-module . Then P ®p Q íz a p ojec í e S-module . P oo . See, o example, Fai h [3, 11 .15, p .4301 . " 2 .2 Theo em . Suppode Q íz a wo-,síded .ideal ín a n .íng R and íe am .Lnímal pnojee í e nígh ideal . Then gí en any pnojee í e n .ígh ideal P o6 R, eí hen PQ = 0 o Q cP . P oo . Since P and Q a e p ojec i e igh ideals and Q is an (R,R)-bimodule, P Q Q is p ojec i e . Sin- ce P is p ojec i e and Q is an idealo R, P ® Q = PQ so PQ is also p ojec i e . Bu PQ c P and PQ  Q . Since Q is a minimal p ojec i e igh ideal ei he PQ  0 o Q= PQ l P . . 2 .3 Co olla y . 1A P .í .6 a m inímal p ojec í e nígh .ideal'_ wh ich .( .6 al'~so a - wo-bíded ídeal, hen eí hen P2 = P on P2 = 0 . " 2 .3-A Rema k . No e ha bo h o he possibili ies men ioned in 2 .3 ac ually occu . Fo example, le R= ¡F 0 lF )F be he ing o 2x2 lowe iangula ma ices o e a ield F . I is well known ha R is semihe edi a y . The ideal P = (o 0) sa is ies P 2 = 0 . Howe e , i R is commu a i e and P is ini ely gene a ed, hen P 2 = P (2 .5) . 2 .4 Rema k . G . Michle [8) has shown ha i P 2 = P in a le pe ec ing R, hen P = ReR whe e e is an idempo en in R and is cen al modulo he adical J . 2 .5 Lemma . In a commu a í e níng R a mínimum pnojec í e .ideal'_ P .C6 ídempoxenx . Ib P ís a16o bíní e1y gene& a ed, hen p i6 gene a ed by an ídempo en el'emen . P oo . Since P is a p ojec i e module, he dual ba- sis lemma gua an ees ha he e exis s a se o elemen s {pa}, pa in P, and a se o homomo phisms { a } wi h a in Hom R (P,R) such ha a (P a ) = 0 o almos al] a, and x = Ep a a (x)  o each  x  in P .  Since P C R,  a commu a i- e ing, we see ha x = E a (p a x) o each x in P . Thus P 2 = 0 implies P = 0 .  Since P 0 0, we see by 2 .3 ha P 2 = P . I P is íni el .y gene a ed, heo em 76, page 50 o Kaplansky [81 says ha P con ains an idempo en ele- men . Since P is a minimum p ojec i e, he idempo en ele men mus gene a e P .a 2 .6 Theo em . 14 Rís a commu a i eeemípn ime n ing wí h EP ha sa ís6íe,6 d .c.c .P,  hen R íe a d inee eum oj a b in i e numben oj b¡eld3 . P oo . Since R sa is ies d .c .c .P, R has minimal p ojec i e  ideals .  Say  1 1  is  a minimal  p ojec i e  ideal . Then  11  is  simple Since  R  has  EP,  and  hence  1 1  is  ge- ne a ed by an idempo en e l . By Jacobson [6, P oposi ion 1,  p .65],  1 1  is a ield .  Le  R 1 =  (1  - e l )R .  Then  R 1 has EP and sa is ies d .c .c .P . Thus R 1 = 0 o con ains a minimal  p ojec i e  1 2  whe e  1 2  is  a  ield  gene a ed  by  an idé iipu en  e 2 .  Then  R 2 =  (1-e1-e2 )R  =  0  o  con ains  a  mi nimal  p ojec i e  13  which  is  gene a ed  by  an  idempo en . Since R has d .c .c .P, R con ains no in ini e di ec sum o p ojec i e ideals so he abo e p ocess mus e mina e a e a ini e  numbe  o  s eps .  Hence  R  =E®I n ,  a  ini e  sum,  whe- e each I n is a ield .a 2 .7 Co olla y . 16 R ís a commu a i e Noe he ian mípn ime n ing zha sa ih6íee d .c .c .P, hen R i6 a dínee bum 06 a 6 in i e numben ob bíeId3 . P oo . Since R is Noe he ian, minimal p ojec i e ideals a e ini ely gene a ed and hence gene a ed by an idempo en . Thus minimal p ojec i es a e summands o he ing . The emainde o he p oo is he same as he p oo o 2 .6 . " We can also say some hing abou he p ojec i e igh ideals in a ing sa is ying igh a .c .c.P . We i s s a e a heo em due o Kaplansky [6] . 2 .8 Theo em . E eny pnojee i e module le a d inec eum o6 coun ablygene a ed modulee . " 2.9 Lemma . 16 R ea el ee n gh a .c .c .P o a n igh d .c .c .P hen e eny pnojee i e n igh Ideal e coun ably gene a a ed . P oo . By Kaplansky's heo em, each p ojec i e ( igh ideal) is a di ec sum o coun ablygene a edp ojec i es . I R sa is ies igh a .c .c .P o igh d .c .c .P he numbe o independen summands o a p ojec i e igh ideal is ini e . Thus each p ojec i e igh ideal is a sum o a ini e numbe o coun ablygene a ed modules a ad hence . i s coun ably gene a- ed . 3 . Inhe i ance P ope ies o Chai a Condi ions o a P ojec i es 3 .1 Theo em . Le R be any n ng . 11 M R sa e6íes he deeeend ng cha in cond i ion o a (6 in i ely gene a ed) pnojee i e 6ubmodules hen each homomonph c mage o6 M aleo sa e 6 eb h e cond on . P oo .  Le  :  M - N  be  a a  R-epimo phism o  igh R-modules a ad le P 1 DP 2 DP 3 D . . . be a sequence o p ojec i e 14 submodules o N . Then -1(P1)---> P 1 = 0 spli s so P 1C -> -1(P 1 ) CM .  Thus P 1 sa is ies he descending chain condi- ion on ( ini ely gene a ed) p ojec i e submodules . Hence he e exis s an n such ha P n = P n+k o k = 1,2,3, . . .- I ollows ha N sa is ies d .c .c . on ( ini ely gene a ed) p ojec i e submodules .a We use he ollowing heo ems o H . Bass [11 and J .E . Bjó k [21 se e a] imes in he p oo o he ollowing esul and in he nex sec ion . 3 .2 Theo em (Bass) . Le R be a n,ing,  J í- z nad .i cal . Then he6ollowíng ane egu .i alen : 1)  R  íz le6 pen6ec ; .L . e . ,  e eny lel R-module haz a pnojec .i e co en . 2) . J .iz lel T-n .ilpo en and R/J .iz zemíz .imple . 3)  A  dínee l .im .i o 6  pno j ee .i e le6 R-modulez  .iz pno j ec .C e . 4) R za íz6íez zhe dezcend .ing ehaín eondí .ion on '  pn .inc .ipal nígh .idealz . 5) R haz no .injín .i e ze a o6 on hogonal .idempo en- z, and e enynonzeno níghz R-module haz nonze o zo- cle .a 3 .3 Theo em (Bjó k) . A níng R .iz le6 pen6ec .i6 and only .i6 R za íz6íez he dezcend .ing ehaín eond .i .ion on 6 .in .i e ly gene a ed nígh ídealz .a * P ojec i eco e is he dual o injec i e hull . 3 .4 Theo em . 16 R ie a nígh penjecz n .n g and M R sazíe6íee a .c.c .( .g)P hen each homomonph ic image o6 M a eo ea íelíes . hís condí íon . P oo . Le : M ->N be an R-epimo phism and suppo- se ha P 1 C P 2 C P 3 C . . . is an ascending chain o p ojec i e submodules o N . Le P = UP i . Then P is la . By Bass [11, P is p ojec i e . Thus -1 (P) - P --> 0 spli s and P embeds in -1 (P) C M . Hence P sa is ies a .c.c .( .g .)P and he abo e sequence o p ojec i e modules mus e mina e a e a ini e numbe o s eps . I ollows ha N sa is ies a .c .c .( .g .)P . " 4 . S udy o Some Pa icula Classes o Ringsunde he Assump ion o Chain Condi ions on P ojec i e Ideals . We can cha ac e ize semip ime igh semihe edi a y ings which sa is y . igh d .c .c . .g.P . Mo e gene ally, a ing in which p incipal igh ideals a e p ojec i e is a níg h  P .P .  níng . 4.1 Lemma .  Le R be a eem ipn .íme n .gh P.P . níng . Then R &a íslíe6 nígh d .c .c . .g .P  í6 and on y í6 R íe e emíeímp e . P oo . Since R is igh P .P ., each p incipal igh ideal is p ojec i e . Thus igh d .c .c . .g . P implies he des cending chain condi ion on p incipal igh ideals . By Bass' heo em, R/J, J = adical R, is semisimple and J, is le T-nilpo en . I J : A 0,  he e exis s a minimal igh ideal I C J .  Now ei he 12 = 0 o he e is an e in I, e : 0, and e2 = e . Since J is le anishing, J con ains no nonze o idempo en s . Thus J = 0 . Hence R is semisimple . The con e se implica ion is clea Since semisimple ings a e A inian . " 4 .2 Co olla y . 16 Ríz a negulan n igh d.c .c . .g .P hen R i3 aemí4 imple . P oo . Regula ings a e semip ime ac , semihe edi a y) and hus by 4 .1 a e sa is y igh d .c .c. .g .P . " n ing sa íz6yíng igh F .P . (in semisimple i hey üe ini ion . A ing R is said o sa is y (a .c .c . .)® i i con ains no in ini e se o independen igh ideals . The a .c .c . on igh annule s o R is abb e ia ed (a .c .c) 1 .  I a ing sa is ies bo h (a .c .c .) ®  and (a .c.c .)l , hen he ing is said o be nígh Goldíe . We say a igh R-module M is uní6onm i X,Y non- ze o  submodules  o  M  implies  X n Y : A 0 .  A  ing  is  (nígh ) un ilonm i R R is uni o m . To p o e ou nex heo em we need he ollowing heo ems o A .W . Goldie [4,51 . 4 .3 Theo em . A n .íng R .íe sem .ípn.íme n .ígh Gold .íe and on1y ,íb .í b quo .íen n .íng Q(R) ib semí .6ímp .Le .a 4 .4 Lemma . A un .íbonm sem,ípn .íme n .ígh Gold .íe n .íng R .íe a n .ígh Une doma .ín . " No e ha in his case, as wi h al] domains, igh a .c .c . .g .P implies a .c .c . on p incipal igh ideals . 4 .5 Theo em . 16 R .íá a n .íng sa .íebyíng n .ígh d .c .c . .g .P, hen R .ís a n,ígh Une n .íng and R = Q(R) . P oo . I a is a egula elemen in R, bu no in- e ible, hen a n R p ope ly con ains a n+1 R o n = 1,2, 3, . . . .  Bu a egula implies  anR- R  is p ojec i e o each n, con adic ing d .c .c . .g .P . Thus each egula elemen in R is igh in e ible . Hence R = Q(R) .@ 4 .6 Theo em . Le R be a 6em .ípn .íme n .ígh . Go .Ld .íe n .íng . Then  R sa ís6íes n .íghx d .c .c . .g .P .íb and onky .íb R  .í6  s em .íb ímple . P o . I R sa is ies d .c .c . .g .P hen R = Q(R) by 4 .5 . Thus by 4 .3, R is semisimple . The con e se is clea .n A domain sa is ying he igh O e condi ion is a igh Goldiedomain . The con e se o his ollows immedia e- ly om he ollowing wo lemmas p o en by A .W . Goldie [4,51 . 4 .7 Lemma .  11  R  sa íes .íes  (a . c .c .) 9  xhen e eny REFERENCES Bass, H ., Fini is icdimension and a homological gene a- liza ion o semip ima y ings, T ans . Ame . Ma h . Soc ., 95, 466-488 (1960) . 2 . Bio k, J . E ., Rings sa is ying a minimum condi ion on p incipal ideals, J . ReineAngew . Ma h . 236, 466-488 (1960) . 3 . Fai h, C ., A .Cgeb&a : Rínge, Module6, and Ca egohíeb I, G undleh en de Ma h . Wiss ., Bd . 190, Sp inge -Ve lag, New Yo k'andBe lin, 1973 . 4 . Goldie, A.W ., The s uc u e o p ime ings unde ascen- ding chain condi ions, P oc . Lond . Ma h . Soc . VIII, 589-608 (1958) . 5 . Goldie, A . W ., Semip ime ings wi h maximun condi ion, P oc . Lond . Ma h . Soc ., 201-220 (1960) . 6 . Jacobson, N ., S huc une og Rínge, ColloquiumPublica ion, Vol . 37 . Ame ican Ma h . Soc ., P o idence 1956 (1964 e i- sed) : 7 . Kapla sky, I ., P o,jec i e Modules, Ann . Ma ii . Ó0 8 . 372-377 (1958) 8 . 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