Lineary compact injective modules and a theorem of Vamos

Pub . Ma . UAB Vol . 30 n2 2-3 Des . 1986 LINEARY COMPACT INJECTIVE MODULES AND A THEOREM OF VAMOS Fo Ka y and Pe e Ca l Fai h 1 A ing R will deno e a commu a i e associa i e ing wi h uni . A e Vámos, R is a SISI Ring i e e y subdi ec ly i educible ac o ing is sel -injec i e . Le M be a maximal ideal o R and le E = E(R/`4)R deno e he injec i e hull o he simple'R-module R/M, and le A(M) deno e he endomo phism ing . Now E is canonically a module o e he local ing RM o R a M, and he unique simple RM -module embeds in E canonically . Mo eo e : E = E ( R/M )R = E(R i/MRCR)RM We call he module (A) he local injec i e hull o R a M, and i s endomo phism ing A(M)  =End E(R/M) R = End E R-  (B) he-local endomo phism ing o R a M . 1 A pa o his pape was w i en sp ing semes e 1986 a he CRM o Ins i u d'Es udis Ca alans o Ba celona, while I was holding a Ru ge s Uni e si y Facul y Academic S udy P og am (FASP) . I ha e he pleasu e o hanking P o esso Pe e Menal o , in i ing me o collabo a e, P o esso Ja mie Moncasi o his many co dial a angemen s on my behal , and o he membe s o he Facul y and s a o helping o make he s ay such happy and ma hema ically p o i able one . 127 R is Vamosi an (classical in [1]) i e e y local injec- i e hull o R is linea ly compac (in he disc e e opology) . In [1] Vamosp o ed ha e e y Vamosian ing is SISI, and ha e e ylocalendomo phism ing o a SISI ing is commu a i- e . We shall p o e he con e se he e, and numbe o subsidia y esul s : 1 . Any SISI chain ing is Vamosian, in ac , an almos maximal alua ion ing (Theo em 9) . As a consequence, we p o e : 2 . A ing R ha is locally a SISI chain ing is Vamos ian (Theo em 10) . Von Neumann egula ings a e locally Noe he ian ings, and a e examples o Vamosian ings ([l]) ; we show ha polyno- mial ingso e hem a e also Vamosian (Theo em 12 and Co olla- y) . A numbe o unsol ed p oblems a e lis ed . One o he main ones asks i R Vámos (SISI) is inhe i edby he polynom- i .a l ing R[x] . This is unknown e en o an almos maximal al- ua ion ing R . 1 . THEOREM . The ollowing a e equi alen condi ionson a ing R : (1) R is SISI . (2) E e y local endomo phism ing o R is commu a i e . (3) E e y R-submodule o e e y local injec i e hull o R is quasi-injec i e . (4) E e y R-submodule o e e y local injec i e module E is an End R E-submodule, i .e . is ullyin a ian . When any o hesehold, hen RM is SISI o e e ymaximal ideal M . Rema k : When his is so, hen e e ylocal endomo phism ing o R is "almos "SISI ; See P oposi ion 4 . Fo he p oo , we need a esul implici in [1] . 2 . PROPOSITION . Le E be an indecomposableinjec i e R-module .  ' The ollowing a e equi alen condi ions . (1) E e y submodule - o E is quasi-injec i e . (2) E e ysubmodule o E is ully in a ian (FI) . (3) E e ycyclicsubmodule o E is quasi-injec i e . (4) E e ycyclicsubmodule o E is FI . (5) I a cyclicmodule R/I embeds in E hen R/I is a sel -injec i e ing . (6)  Fo each a EA = End ER and x E E,  he e exis s E R such ha a(x) = x . When his is so, hen A = End RE is commu a i e . PROOF .  (1) < * - (2)  by a heo emo Johnson and Wong ([7], p .63 . Co 19 .3), which s a es ha an R-module M is quasi-injec i e i M is ully in a ian in E(M R ) . I e e y cyclicsubmodule o  E  is ully-in a ian (quasi-injec i e) hen e e y submodule is, hence (1) - (4) a e equi alen . Ob iously (4) - (6) . Fu he mo e (5) - (3), because R/I sel -injec i e implies R/I is quasi-injec i e (since e - e y R-submodule is an R/I-submodule) . (3) =»(5) . I R/I is quasi-injec i e qua R-module, i is quasi-injec i e qua R/I-module, equi alen ly, sel - injec i e  by Bae 's c i e ion  ([6], p .157, Theo em 3 .41 .) E iden ly, (6) implies ha A is commu a i e . 3 . COROLLARY . (Vámos) A ing R is SISI i e e y local injec i e module  E = E(R/M) R  sa is ies any o he equi alen condi ionso he p oposi ion . PROOF . This ollows since e e ysubdi ec ly i educible ac o ing  R/I  embeds in  E(R/M) R ,  whe e  R/M = socle R/I ; and con e sely, i  R/I - E(R/M), whe e  M  is maximal, hen R/I is subdi ec ly i educible . (1) ° (2) by Vámos [1], and (1) « (3) by Co olla y 3 . Mo eo e in iew o (A) and (B), Co olla y 3 also yields (3) - (4) . (2)  (1) . I su ices o p o e ha R M is SISI o e e y maximal ideal M, hencesuppose R is local wi h maximal ideal  M .  In his case  E = E(R/M) R  is an  injec i e cogene a o o mod-R . Nowle M = R/I be a subdi ec ly i educible ac o ing, andle E = E(M R ) be i s injec i e hull aken in E . PROOF OF THEOREM 1 . This can be donebecause E is alsoessen ialo e M as an R-module, and  E = E( ;_1 R ), since M --4 - E and E is indecom- posable . Fu he mo e M = ann E I D E, and M is essen ial o e M as an R, whence as an R-module, so he e o e M = E . This shows ha E is a ully in a ian submodule o E since ob iously whe e K = annAE . so ha aM CM V aE A .  Thus The ac ha M ---> E implies ha E is a cyclic P .-module (e .g ., see [8], p .15, P op . 5 .5), whence E ^A in mod-A .- Fu he mo e, Q = Qmax(R)C- A A = A/K ^ End ER E ^A^ Biend El = Q  (C) canonically (e .g . [7], p .81, P op . 19 .21 .) . Since ER is injec i e, hen i is quasi-injec i e o e Q = Biend ER by Co olla y 5 .6A, p .15 o [8) . Using (C) we see ha A is sel -injec i e . We also need he ac ha A is an injec i e cogene a o o e R (since E is) . Thus e : e y ideal  H  o  R  is he annihila o o an .A-submodule o E  A , . say H = annIG o an ideal G o A . (See [7], p . 190, Co olla y 23 .23 .) Bu i H is chosen o be a dense ideal o R (= R is a a ionalex ensiono  H),  hen he only ideal in A = Qmax(R) ha annihila es i is ze o . See, e .g . [71, p .80, 19 .32(b), which implies ha ann-annAH = E o any denseideal H o R . In ou con ex , his means ha ann A H = 0, so G = 0 whence H = R . Bu , hen R= A, since o e e y q E Q he exis s a dense ideal I o R wi h qI --~ R . This p o es ha R is sel -injec i e, and hence R is SISI . Rema k : (1) A p oo o (2) - (1) has kindlybeenp o idedby P o esso Vámos, who also supplied he ollowingexample (2) o a locallyVamosian ing ha is no SISI . Suppose (2) holds bu (1) ails . As be o e, we may assume ha R is a local subdi ec ly i educible ing embedded in E = E(R) . Since E >,¿ R he eexis s xE E R, and since E is an injec i e cogene a o , he e exis homomo phisms such ha Pa iallyDi e en P oo E --> E and R : E --~ E a (1)  = x,  Q (1)  = 0,  0(x)  y, 0 . Then 0~ G (x)  =  a (1)  76 Cío (1)  = 0 con adic ing commu a i i y o End RE . (2) Le R be a subdi ec ly i educiblealmos maximal o ch ing, i .e . a ing R wi h : (i) a leas wo maximal ideals such ha R M is an almos maximal alua ion ing o each ME max R, (ii) a wais P, whe e P is a minimalp ime and a unise ial module, and such ha R/P is an h-local do- main [i .e . e e y nonze o p ime is con ainedin a unique maximal ideal, and e e ynonze oidealo R/P is con ained in jus ini ely manymaximal ideals .] Such ingsexis (see [15]) bu canno be SISI since R is no sel -injec i e . (An indecompos- ablesel -injec i e ing is local .) The nex esul shows ha a localendomo phism ing o a SISI ing is "almos " SISI . 4 . PROPOSITION . I E is an injec i e R-modulewi h commu a i e endomo phism ing A, any A-submodule o E is quasi-injec i e, and A modulo any ideal I such ha A/I C- . E  is sel -injec i e ; equi alen ly  A/I  is sel injec - i e o any ideal I = ann A x o some x E E . PROo . As s a ed in he p oo o Theo em 1, E A is quasi-injec i e,and A = End AE .  I S C M a e A-submodules o E, and : S + M an A-map, hen by quasi-injec i i y o E o e A, is induced by a EA . Since M is an A-submodule,  a MC M,  hence ex ends o an endomo phism o . M A . This p o es quasi-injec i i y o any A-submodule M o E . The sel -injec i i y o A/I ollows om'i s quasi-injec- i i y as in he p oo o P oposi ion 2 . I : A/I -> E is an embedding o A-modules, hen I = ann A x, whe e x = (1+I) . Con e sely, i I = ann Ax hen he e is an embedding A/I ----> E sending  A + I - ax G a EA . No e, i R is SISI, hen e e y local endomo phism ing, A = End E(R/M) R , is commu a i e, and he uniquesimple A-mod- ule W embeds in E and coincideswi h V .= R/M . Thus, he p oposi ion would imply ha A is SISI p o idedonly ha E is injec i eo e A . This is no in gene al ue o a SISI ing R . In ac , Vámossingles ou a classo ings (called classical in [1J) o ec i y his de iciency . We say ha . R is a Vámos ing , o Vamosian ( o me ly classical) p o ided ha e e ylocalinjec i e hull is linea ly compac (l .c .) o e R . We employ he e minology injec i endo o indica e when a module F o e R is injec i eo e i s endomo phism ing A . An ideal I is co-subdi ec ly i educi- ble (co-SDI) i R/I is a subdi ec ly i educible ing . VÁMOS THEOREM [1] . I R is Vamosian hen : (V1) R is SISI . (V2) The local endomo phism ing A a any maximal ideal M is he .comple ion o R M in he opology gene a ed by he co-SDI ideals o R M , and is a l .c . ing . (V3) E e ylocalinjec i e hull E is injec i endo, and l .c . o e i s endomo phism ing A, equi a- len ly Hom A ( E) induces a Mo i a duali y in mod-A (on he ull subca ego y o l .c . A-modules) . (V3) Follows om heo ems o Mo i a [4)  and Muelle [3], which imply ha a commu a i e ing A has a Mo i a dua- li y i he leas injec i e cogene a o E o e A sa is ies A = End A E . By Muelle [3] his is equi alen o equi ing ha bo h A and E be l.c . A-modules . 5 . THEOREM . The ollowing a e equi alen dondi ions on a ing R (1) R is Vamosian . (-2) R is SISI and e e y local endomo phism ing is Vamosian . (3) R is SISI and e e ylocalinjec i emodule is injec i endo . PROOF .  (1) - (2) . By Vamos' heo em, R is SISI, and e e ylocal endomo phism ing A = End ER has l .c . . injec i e hull E by (V3) . (2) = *»(1) . Le E be a local injec i emodule o R, and A = End RE . Since A is Vamosian, hen he injec i e hull F o i s uniquesimplemodule W is l .c . o e A . Bu , W C--, E  and,  in ac , coincides wi h he uniquesimple R-mod- 11 LEMMA . I P is a p ime ideal o R[x], and P0 is he con ac ed ideal in R, hen R[ x] P ^ Rp 0 [x]Pex whe e  E ex  is he ex ension o  P  o  R P [x](i .e .-Pex=PR P [x]) . O  O PROOF .  P ex  consis s o all  g(x)  in  Rp [x]  wi h 0 coe icien s in  PR P ,  and  P ex  is p ime since,  in gene al, 0 o any ing A and p ime ideal L o A, we ha e A[x]/L[x] - A/L[x] is a domain . Le (x) = h(x)/g(x) deno e an elemen o he igh side, i .e . Le h(x), g(x) E Rp [x], wi h g(x) 9P ex . We can 0 w i e .h(x) =h0 (x)/c  and  g(x) = g0(x)/d wi h c,d E R P 0 , and g 0 (x), h b (x) E R[x] . Since c,d ¢ P0, hen -cdg0 q! P, hence h(x) =h0(x)/cdg .0(x) E R[x] p . The e e se inclusion is  p o ed simila ly, i .e ., i h, g E R[x], and g ¢ P, hen we may iew h and g as elemen s o R p [x], 0 and .mo eo e , g ¢ P ex m so h/g E R p [x] ex U P 12 . THEOREM . . I R is locally Noe he ian, hen so is any polynomial ingo e R in ini elymany a iables x 1 ,.. .,x n . In pa icula , hen R[x1, . . .,xn] is Vamosian . PROOF . Since  R  is locally Noe he ian, hen  R P0 [x] is Noe he ian o any p imeideal P o R[x], and hence, by Lemma 13, so is he local inga P . 13 . COROLLARY . I R is on Neumann egula , hen R[x1, . . .,xn1 is Vamosian . 14 . REMARK . R[x] is hen semihe edi a y, and con e se- ly, i he polynomial ing R[x] o e a (no necessa ily commu- a i e) ing is semihe edi a y, hen R mus be on Neumann egula . (See [12, 13, and 14] . (Howe e ~gene al, on Neumann egula ing R does no imply R[x] semihe edi a y) . 15 PROPOSITION . I R is Vamosian ( esp . SISI), hen so is e e y ac o ing . PROOF . I R is SISI, hen e e y ac o ing ob iously is, so suppose ha R -is Vamosian, and I is an ideal, and V a simple R/I module, and E he injec i ehullo V in mod-R . I is easy o see ha he annihila o E o I in E is he injec i e hull o V in mod-R/I (c . he p oo o (2) =* (1) o Theo em 1) . I ollows ha É is l .c . o e R/I, since E is l.c . o e R, hence R/I is alsoVamosian . is R . 16 . COROLLARY . I R[x] is Vamosian(SISI), hen so R[x] is monic i i con ains a monic polynomial An ideal I o R[x]  is monic i I con ains a monicpólynomial, equi alen ly, R[xl/I is a ini ely gene a ed R-modulé . A ing R is called a Mo nica ing i e e y co-subdi ec lyi educible ideal o R[x] is monic . An ideal I o R is colocal i R/I is a local ing . Example . Any co-SDI ideal I o a SISI ing is colocal, since R/I is hen indecomposable injec i e, hence has local endomo phisin ing which is isomo phic o R/I . In his example I is also co-PF in he sense ha R/I is PF . Thus, R/I has a Mo i a duali y, and hence R/I is Vamos . I P is a subca ego y o he ca ego y RINGS, hen o any ideal H o a ing A, we say ha H is a co-P-ideal i A/H E P . In his pape in e aliawe ha ebeenin e es ed in subca ego ies o RINGSconsis ing o : i educible (l .e . uni o m) ings, local ings, semilocal ings, semipe ec ings, sel - injec i e ings, PF- ings, and (locally) Noe he ian ings . 17 . Theo em . I R is l .c ., hene e ymonicideal I o R[x] is co-semipe ec , i .e . o co-local ideals Ii D I, i =  and > 1 . Consequen ly, any monic co-i educible idealo R[x] is co-local . R[ x1 /I = R[X]/I1x . . . XR[ x] /I P oo . Since I is monic, hen R[x]/I is ini ely gene a ed o e R, and hence by [1] o [3], is 1 .c . as an R-module . By [161, any l .c . ing is semipe ec , so R[x]/I has he s a ed decomposi ion . 18 . THEOREM (VAMOS [2]) . I R is a Mo i a ing (i .e ., has a Mo i a duali y), hen so does any algeb a A o e R ha is l.c . o e R, in pa icula , ha is a ini ely gene a ed R-module . 19 .COROLLARY . I R is a Mo i a ing, hen R[x]/I is a Mo i a ing o any monicideal I, and hence R[x]/I is sel -injec i e o any monic co-SDI ideal I . P oo . Ob ious om he aboye heo em o Vamos and he p oo o Theo em 17 . 20 . COROLLARY . I R is a MonicaMo i a ing, hen R[x] is SISI . P oo . By Co olla y 19, R[x]/I is Mo i a,hence Vamos, and he e o e SISI, o e e y co-SDI ideal . 21 . COROLLARY . I R is a l.c . VamosianMonica ing, hen R[x] is SISI . P oo . A ing R is Mo i a i R is l.c . and Vamosian,acco ding o Muelle 'sTheo em s a ed ea lie so R x  is SISI by Co olla y 20 . 22 . PROPOSITION . I R is a l .c . ing, hen he Muelle -Vamos conjec u e implies ha R[x]/I is Mo i a o any monic ideal I . P oo . Ob ious, since R[x]/I is a ini ely gene a ed moduleo e he Mo i a ing R . 23 . COROLLARY . I R is a l .c . Monica ing, hen MVC implies ha R[x] is SISI . Monica? 25 . REMARK . I R is SISI, i I is a co-SDI idealo R[x]  and i I n R is a co-SDI ideal o R, hen I can show ha I is monic i I is co-local . This esul will appea elsewhe e . P oo . Clea om he p oo o Co ola y 21 . 24 . QUESTION . Is e e yMo i a,Vámosian, o SISI ing REFERENCES 1 . Vámos, P ., Classical ings, J . Algeb a 34 (1975) 114-129 . 2 . ---------, Rings wi h duali y, P oc . London Ma h . Soc . 35 (1977) 275-289 . 3 . Muelle , B ., Linea compac ness and Mo i a duali y, J . Álge- b a 16 (1970) 60-66 . 4 . Mo i a, K ., Duali y o modules and i s applica ions o he heo y o ings wi h minimum condi ion,Sci . Repo s, TokyoKyoiku Daigaku 6 (1959) 83-142 . 5 . Ma lis, E ., Injec i emoduleso e Noe he ian ings, Pac . J . 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