The Lattice of Fully Invariant Subgroups of the Cotorsion Hull

Transcription

1 Advace Pure Mahemac Publhed Ole November 3 (h://wwwcrorg/oural/am) h://dxdoorg/436/am3389 he Lace of Fully Ivara Subgrou of he Cooro Hull arel Kemoldze Dearme of Mahemac Aa ereel Sae Uvery Kua Georga Emal: emoldze@gmalcom Receved Seember 9 3; reved Seember 8 3; acceed Ocober 7 3 Coyrgh 3 arel Kemoldze h a oe acce arcle drbued uder he Creave Commo Arbuo Lcee whch erm urerced ue drbuo ad reroduco ay medum rovded he orgal wor roerly ced ABSRAC he aer coder he lace of fully vara ubgrou of he cooro hull whe a earable rmary grou a arbrary drec um of oro-comlee grouhe vegao of h roblem he cae of a cooro hull mora becaue edomorhm h cla of grou are comleely defed by her aco o he oro ar ad for mxed grou he rg of edomorhm omorhc o he rg of edomorhm of he oro ar f ad oly f he grou a fully vara ubgrou of he cooro hull of oro ar I he codered cae he cooro hull o fully rave ad hece eceary o roduce a ew fuco whch dffer from a dcaor ad ag a fe marx o each eleme of he cooro hull he relao dfed o he e of hee marce dffere from he relao rooed by he auor he couable cae ad beer dcrbe he lower emlace he ue of he relao eeally mlfe he verfcao of he requred roere I roved ha he lace of fully vara ubgrou of he grou omorhc o he lace of fler of he lower emlace Keyword: Lace of Fully Ivara Subgrou; Drec Sum of oro-comlee Grou; Cooro Hull Iroduco We coder queo of he heory of abela grou ad hroughou he aer he word grou mea a addvely wre abela grou he oao ad ermology ued he ex are borrowed from he wovolume moograh [] he ymbol deoe a fxed rme umber Z ad Q are reecvely he grou of eger ad raoal umber If he order of a eleme a of he grou he he exoe of a eleme a equal o ad wre a ea A ubgrou B of he grou A called fully vara f for ay edomorhm of he grou A h ubgrou B maed o B he examle of uch ubgrou are A a a A A a a aa Z he oro ar of he grou A he udy of he lace of fully vara ubgrou of a grou a mora roblem of he heory of abela grou For uffcely wde clae of -grou h oc reaed [3-7] ad oher aer he wor [8-3] ad oher are dedcaed o he vegao of h queo oro-free ad mxed grou A grou A called a cooro grou f exeo by mea of ay oro-free grou C l e Ex C A he morace of he cla of cooro grou he heory of abela grou due o wo fac (ee [ em 54 55]): for ay grou A B he grou Ex A B a cooro grou; ay reduced grou A omorhcally embeddable o he grou A Ex Q Z A called he cooro hull of a grou A ad addo A A a dvble oro-free grou Ay reduced cooro grou A ca be rereeed a he drec um A C where Ex Q Z = A he oro ar of a grou A ad C a oro-free algebracally comac grou (ee [4]) If rereeed a a drec um of rmary comoe he Ex Ex Q Z Z he coruco of algebracally comac grou well ow (ee [4 em 4]) Hece he udy of cooro grou reduce he udy of grou of he form Oe Acce

2 KEMOKLIDZE 67 Ex Z where a -rmary grou o a coderable exe hough he oo of a cooro grou ad geeralzao are uded uffcely well (ee [5-8]) lle ow abou he lace of fully vara ubgrou of a cooro grou he vegao of h roblem he cae of a cooro hull mora becaue edomorhm h cla of grou are comleely defed by her aco o he oro ar ad a how [9] for mxed grou he rg of edomorhm omorhc o he rg of edomorhm of he oro ar f ad oly f he grou a fully vara ubgrou of he cooro hull of oro ar he udy of he lace of fully vara ubgrou mae eeal ue of he oo of a dcaor ad a fully rave grou By he -dcaor of a eleme a of he grou A we mea a creag equece of ordal umber HA a Ha ha ha h a where h deoe he geeralzed -hegh of he eleme a e ha f a A\ A ad h (ceraly f h a h he h a) I he e of dcaor we ca roduce he order H a H b h a h b A reduced -grou called fully rave f for arbrary eleme a ad b whe H a Hb here ex a edomorhm of he grou uch ha a b I fully rave grou he lace of fully vara ubgrou uded by mea of dcaor (ee [ heorem 67]) A Mader [] howed ha a algebracally comac grou fully rave ad decrbed by mea of dcaor of he lace of fully vara ubgrou of a algebracally comac grou Moreover he dcaed he geeralzed codo he fulfllme of whch gve a decro of he lace of fully vara ubmodule heorem (A Mader) Le A be a module over a commuave rg R be he lace of fully vara ubmodule be ome lower emlace ad : A be he mag wh he followg roere: ) urecve; ) faa aa ad f Ed A; 3) abab; 4) f ab he here ex a edomorhm f of he module A uch ha f b a; 5) f C he for ay ab C here ex cc uch ha c ab he he e of all fler of whch ordered wh reec o he cluo a lace ad he mag : defed by he rule D a A a D a lace omorhm I he ame way a we dd -grou we defe he oo of full ravy he grou Ex Z If a oro-comlee grou he cooro hull a algebracally comac grou (ee [ em 56] ad a ha bee meoed above fully rave A Moaleo [3] roved ha whe he drec um of cyclc -grou he alo fully rave ad all he codo of heorem are fulflled herefore h cae oo he lace of dcaor fler decrbe he lace of fully vara ubgrou he drec um of oro-comlee grou a aural geeralzao of he drec um of cyclc -grou ad oro-comlee grou he auhor ha how [] ha h cla of grou f he um fe he cooro hull o fully rave herefore becaue of codo 4) of heorem we cao ue dcaor o decrbe he lace of fully vara ubgrou he lace of fully vara ubgrou of wa uded [] whe he couable drec um of oro-comlee grou I he ree aer a arbrary drec um of orocomlee grou ad he lower emlace defed by a mler ew relao (ee Defo ) whch mae eaer o verfy he roere of heorem A Semlace Le a -grou be he drec um of orocomlee -grou B () where B B B a bac ubgrou of he grou B ad he bac ubgrou of Aume ha a fully ordered e of dexe For a earable -grou A Moaleo [3] rereeed eleme of he cooro hull a couable equece a a a a a a Wrg he eleme h form eay o calculae her hegh ad dcaor (ee [] ()) Le B x be a fxed bac ubgrou of a J earable -grou If a a a a he he grou B here ex a equece b uch ha for ay m x m a b = b ad lm () h rereeao of a eleme a called Oe Acce

3 67 KEMOKLIDZE caocal he equece b ad o correod o he caocal rereeao of a he aeme gve below are rue (ee [3]) Prooo If H a b he equece correodg o he caocal rereeao of a eleme a ad bewee ad here a um he he exao b wh reec o he ba x here a eleme x of order Prooo If H a a equece of oegave eger he ha a fe umber of um Le a grou be of form () ad a Deoe by he roeco of he grou o he drec ummad B ad coder he equece b b (ee ()) For each ad fxed he equece b b defe he eleme a = lm b (3) ad he eleme a a of he grou B defe he eleme a a a a of he grou I obvou ha a lm (4) a (Here we have earaed wo dexe ad by he comma ad we wll omeme do o he equel order o dguh her order) Noe ha he eleme a are uquely defed by a eleme a (ee [3] em ) o every eleme a we u o correodece he marx a Ha H a (5) where H a H a H a for ad he dcaor are wre a colum Defo he marx made u of ordal umber ad ymbol called admble wh reec o he grou f he followg codo are fulflled: ) he h row a creag equece of ordal umber o ha or If he for ay wherea he oher row are creag equece of oegave eger umber or ymbol (Here he malle fe ordal umber ad aumed ha ) ) If m he fr fe ordal umber ad m he fely may row coa a oegave eger umber ad here ex a row uch ha m for If m m he arg from ome all row co oly of ymbol 3) If all eleme a row are oegave eger he h row coa fely may um 4) If bewee ad here a um he he grou B here ex a bae eleme of order ( aumed ha B B) 5) I each colum a (e ru hrough all value of ay fully comleely ordered e ); alo f m he m ad f m he ag o accou equaly () ad Prooo we oce ha he marx a afe he above codo for ay a From Defo follow ha we deal wh marce of he followg hree ye: where ; II I are oegave eger umber or ymbol m m where m ad are oegave eger umber (ee he fr eece em of Defo ); III Here are oegave eger umber (ee he ecod eece em of Defo ) Noe ha f a a a gve ad b he equece correodg o a caocal rereeao of a eleme a he a b b b ad each b he um of fely may bae eleme x (ee ()) Hece follow ha ay marx a a mo a couable umber of row dffere from Moreover by vrue of he fourh codo of Defo from he gve h row of a admble marx eay o fd a eleme a he dcaor of whch equal o h gve row Le for examle Oe Acce

4 KEMOKLIDZE (6) ( obvou ha here ad he dexe are mared order o deerme whch row ad colum a eleme le) Jum here occur a oo he by vrue of he fourh codo of Defo (ee alo Prooo ) he caocal rereeao of a here he eleme b3 b coag a ba eleme of order b b Ju he ame way coa a bac eleme of order ; b coa a eleme of order b 4 ; b a eleme of order b 6 a eleme of order b 8 b 7 ad o o herefore a b b b b b he dcaor H a obvouly equal o equece (6) ad a o he oly eleme he dcaor of whch equal o (6) Deoe by he e of admble (wh reec o ) marce ad defe o he e he followg relao dffere from he relao gve [ Defo ] Defo Le K K be he eleme of he e We ay ha K K f ad o each eleme where here occur a um we ca u o correodece he eleme m where here alo occur a um o ha ad he he followg wo codo are fulflled: ) Each eleme m where here a um ha fely may (obly oe) re-mage ) If are fely may eleme of he h row of he marx K whch are dffere from he ymbol ad m m are reecvely her re-mage uch ha he equece of he umber of row m m fely creae he a I ca be ealy verfed ha he relao o he e reflexve ad rave However a ee from he ex mle examle o a-ymmerc Ideed le U u V v be admble marce all row of whch exce for he hrd oe are decal Coder he able m m u u u v v v u v u v u v u v u v Le u aume ha he eleme lyg a he oo of do he marce U ad V are decal o he eleme of he hrd row of he marx U where here are um we u o correodece he eleme of he ecod row of he marx V hu o he eleme of he hrd row of he marx V where here are um here correod he eleme of he hrd row of he marx U he followg maer: I obvou ha U V ad V U wherea U V herefore he relao o he e o a-ymmerc he UV U V ad V U he relao of equvalece o he e def U V U V def wherea defed o he facor e he relao of order Le U u V v be admble marce We deoe W U V m u v w ad wll how ha W alo a admble marx Le u u (7) v v (8) Oe Acce

5 674 KEMOKLIDZE be reecvely he h row of he marce U ad V where here occur fely may um Le u how ha he m u v m u v (9) alo a creag equece of oegave eger umber where here are fely may um Aume he corary: le arg from ome umber (9) here are o um ad w u v Aume ha u u he fr um o he rgh from u (7) ad vm v m he fr um o he rgh from v equece (8) ad m he w u w u w u ad obvouly w v If m he wm m ad wm u vm whch coradc he defo of W herefore he hrd codo of Defo fulflled Le w w be a um (9) ad w u he u w wu ad u a um (7) he B u u Bw e he fourh requreme of Defo alo fulflled he fulfllme of he remag codo of Defo obvou herefore W a admble marx I o dffcul o verfy ha W U ad W V Moreover f K U ad K V he K W Le ow UV where U ad V are admble marce Le u defe he exac lower boud of U ad V a follow: f UV UV W where W m u v If U u U ad V v V he by vrue of he above roere W m u v UU ad WVV Hece W W ad by ymmery W W e W = W whch how ha he defo of W = U V reaoable Sce W U W V we have W U W V ad f K U K V we have K U K V K W e K W hu all codo of he defo of he exac lower boud are fulflled herefore he e wh relao he lower emlace 3 he Lace of Fully Ivara Subgrou of he Grou Le u how ha he fuco : aa where ha form () ad he e of all admble marce wh reec o afe all codo of heorem Codo urecve Proof Le K K = ad he h row co of oegave eger umber For ay h row deoe all um by By vrue of he admbly of he marx K for each um of h d we ca chooe he ba x of he grou B a eleme x of order Le b Deoe x x c x he a lm c ; a B ag o accou ha ad for every we ee ha H a Furher ce B ad he marx admble for each fxed we have a Now we ca defe he eleme a l a H a ad a Sce a dvble grou here are eleme a a uch ha for ay we have a a Le a m a a a he a ad K Now aume ha he marx K of ye II he h row ha he form m m a K where m ad here ex a dex uch ha m for every Le he recedg cae for each -h row here ex a eleme x a x x B Here he umber of ummad fe ce he row ha fely may um Now ag o accou ha ad he marx K admble we oba a he we ca defe he eleme lm a x x x ad H a B Deoe lm r whe l r a x x x aumg ha ha x r he obvou ha a a Coder a eleme a a a I ealy follow ha a K herefore a K If K a marx of form III he arg from ome -h row every row co oly of ymbol We chooe a row ad u he ame way a he recedg cae fd a x x x B ad H a Le u coder a B Oe Acce

6 KEMOKLIDZE 675 eleme a x x a he H a Sce a dvble grou ad here ex a qua-cyclc dvble ubgrou Le g be yem of geeraor uch ha g g g ad for every Sce a ure ubgrou ca be aumed ha g for each Now le a a a a g ad coder he eleme aa a a a I obvou ha a K e a K Codo roved Codo If a ad f Ed he a f a Proof Le aa a cc c ad here ex a edomorhm f of he grou uch ha f a c A ow (ee [ em 5]) f duced by he edomorhm f of he grou whch ur duce a edomorhm f of he algebracally comac grou uch ha f a fa fa c c c b Le ad d be reecvely he equece correodg o caocal rereeao of he eleme a ad c he (ee (3) (4)) lm a a a b where ru hrough a mo a couable e of creag dexe Sce a algebracally comac grou we have fa fa c c (3) Le a c ad aume ha here a um a he oo he a comoe c of he eleme c here ex a eleme d of he exoe whch by vrue of equaly (3) obaed by mag he um of a fe umber of ummad of he eleme a uder he homomorhm f ad by roecg h mag o a ba eleme of he ubgrou B ha ha he exoe Deog h roeco by we have f b b (3) d where he above-meoed um of a fe umber of ummad ecloed he brace I obvou ha he hegh of uch a ummad le ha or equal o wherea he exoe greaer ha or equal o Hece whou lo of geeraly we u o correodece o a eleme of he large exoe hu o each eleme of he marx K where here a um we u o correodece a eleme of he marx K Le be eleme of a row of he marx K where here are um ad m be m reecvely her remage for he above-meoed correodece o ha a equece of umber of he row m m of he marx K creae fely ag o accou ha a algebracally comac grou ad oro ar a oro-comlee grou by vrue of equaly (3) we aume whou lo of geeraly ha m f b d m m m m We have m m m m f b b m m m m m m d d (33) Sce f duce a edomorhm o ad are roeco m m m m m m f duce a edomorhm o he ubgrou Le u fx a ove eger umber m ad coder a eleme of m order from : m m a b b If here he al ummad m he we aume ha hee ummad are zero he vew of (33) m m m m m m m m m m a b b d d (34) Sce by codo a equece of umber of he row m m of he marx K creae fely oly a fe umber of ummad o he rgh-had ar of equaly (34) mu dffer from zero; oherwe he eleme a doe o belog o herefore for each cocree ove eger m arg from ome we have m m e a hu he ecod codo of Defo fulflled oo herefore a f a ad Oe Acce

7 676 KEMOKLIDZE a f a Codo roved Codo 3 For ay ac aca c Proof Le a a a c c c be he eleme of he grou he (ee [3 em ]) aca c a c Deoe (ee (5)) a H a c H c ac ac H a c By vrue of he roere of he dcaor (ee [ em 37]) Ha c HaHc for ay Le ad a here be a um he If o he rgh from he fr um occur a he oo he ad o he eleme we u o correodece he eleme For h correodece f are he eleme of he h row where here occur um he her re-mage le he ame h row of he marx ad herefore he ecod codo of Defo wll be fulflled oo e aca c or aca c Codo 3 roved Codo 4 If ac ad a c he here ex a edomorhm of he grou uch ha a c Proof Le aa a cc c ad b a lm b c lm d where d are reecvely he equece correodg o her caocal rereeao ) Deoe a c ad aume ha he equece coa oly oegave eger umber Le be he eleme of he -row whch have re-mage he marx c Deoe he re-mage of he eleme a follow (35) m m he he eleme a ha he ummad b where b coa a a ummad he ba eleme x x of he exoe Deoe A x m m m m We wll how ha for each here ex a edomorhm of he grou uch ha b b b m d d d m m (36) ad x for ay bac eleme x coaed he equece b of he caocal rereeao of a whe x A Noe ha whe he ummad of he eleme b b b do o coa he ba eleme x whe Ideed whe we have Hece (37) h a O he oher had ad f we ae o accou he defo of he hegh ad he rereeao of he eleme a he for each m we wll have b herefore by m (37) b b bu for each e x Deoe by m he coeffce wh whch x coaed he exao of he eleme b Recall ha m he codo b b b m m m x m d d d m m (38) mu be fulflled for he ough homomorhm Sce x arcae he exao b wh reec o he ba we have m herefore x ca be uquely defed he ubgrou B from (38) f e x (39) Bu h equaly hold rue becaue he re-mage of are (35) (for ) ad by he defo of he relao bewee he marce a ad c for each m we have d d d Oe Acce

8 KEMOKLIDZE 677 ce whch rove he valdy of equaly (39) Ju he ame way a he cae we wll how ha he exao of eleme b b b 3 here o x whe herefore he codo b b b m m m x 3 3 m m m x m m m d d d fulflled for he edomorhm Sce m accordg o h codo we defe x uquely Ju le for x we ca verfy ha e x ad o o he edomorhm lewe defed uquely by gvg he mage of ba eleme ad obvou ha ma he bac ubgrou o he edomorhm uquely coue u o he edomorhm of he grou Le u how ha duce a edomorhm o he grou Le Sce he grou ha form () uffce o how ha whe B Sce for he ba eleme x we have x whe x A ug (36) le whou lo of geeraly be a eleme of m order m m b b b b b b (Here aumed ha e b ad f m everal ummad m he hee ummad are equaed o zero) he m m b b m m b b m m b b m q d d m m d q d m m m d d m m (3) where q If equaly (3) he umber of row m of he marx () c fely creae by vrue of Defo (ee alo (35)) m whe fely creae Bu he m m m m m m m m m m ' ed m m m m h mea ha arg from ome all ummad o he rgh-had de of equaly (3) are equal o zero herefore he um of edomorhm whch o he algebracally comac grou duced by he edomorhm o he h comoe of he grou B he edomorhm of he grou whch ma he ubgrou o I ca be ealy verfed ha a c ur defe he edomorhm of he grou for whch c a a a c c A he begg of he roof we have aumed ha he h row H a co of o- egave eger umber ad he h row here are a fe umber of um I obvou ha h reaog alo rue whe he h row coa a fe umber of um (a lea oe um) or whe he h row of he marx a ha he form H a m m ad c c c ) Le u earaely coder he cae where H a m m or ce ure aume ha he oao a a a we have a a a m ad am a he he rereeao m m a lm b b coa a ba eleme m x B of arbrarly large order ad ae value from a fe e of dexe Le u fx a equece of ove eger umber uch ha he exao of b coa a ba eleme x B uch ha e x m Le Ax We aume ha x for x A ad aalogouly o ar of he roof for ay we defe x uch ha he equale Oe Acce

9 678 KEMOKLIDZE b b b d d d (3) are fulflled I ca be aumed ha m Noe ha x doe o arcae he exao b wh reec o he ba whe Ideed f were o o he he exoe of he eleme m b m b m b would be larger ha whch o o eam m m Moreover e x e x Ideed whe a corme umber wh reec o erve a he coeffce x herefore uffce o m how ha d d We d have bu c d d d m m m m m d c d d d m m m m m d Hece ce dm o dvble by all ummad mu be equal o zero e ex x Aalogouly e x ex ad o o I obvou ha by vrue of (3) x ad by our coruco x belog o varou B ad herefore duce he edomorhm o he grou ad ma he ubgrou o For he duced edomorhm o he grou we have a a a m m c c m 3) Le H a m m he H a m m H c ad a how ar of he roof here ex a edomorhm f of he grou whch duce he edomorhm o he o ha f a c Hece f ac e f ac ad H H f ac H a herefore by vrue he la eece a he ed of ar of he roof here ex a edomorhm of he grou whch duce he edomorhm o he ubgrou o ha a he f ac a or f a c Obvouly f duce he edomorhm o ad a c Codo 4 roved Codo 5 If C a fully vara ubgrou of he grou ad ab C he here ex c C uch ha c ab Proof Le ad l l be reecvely he h row of he marce a ad b Le be a malle dex uch ha l I hee equece he eare um o he rgh from Le h um occur a he oo he equece I he laer equece o he lef from here he recedg um Le h um occur a he oo he from o cluve we add m o each eleme o ha bewee m ad here would be o um Obvouly m m (3) exceed reecvely l l ad o he lef from he dex each l he obaed equece ad where ad dffer from by he eleme of (3) obvouly afy he codo of a row of he admble marx ad Now le u aume ha he equece ad he equaly of eleme ae lace a he umber l where he hee equece we have o he rgh from a um ad reea he revou reaog If he equece here are fely may um ad a each age he fr um occur oe ad he ame equece r he o o volae codo 3 of Defo we roceed a follow: le he equece r ad r a he oo l ad o he rgh from here occur a um bewee lm lm he he fr equece r where here are fely may um here ex a um uch ha l l O he rgh from l we creae he umber l lm o ha here would be o um bewee he umber l m lm e he equece r we ourelve have eoally creaed a um bewee he umber l ad l Noe ha a h oo he codo of admbly of a row ha o bee volaed ce l ad here ex a um bewee ad We have ; Deoe he row ad are admble ad where each eleme of dffer from he correodg eleme of Now f every row of he marce a ad b we erform uch raformao he we oba admble marce U ad V he correodg eleme of whch dffer from oe aoher ad a U b V a b U V I o dffcul o verfy ha h reaog hold for all ye of marce a ad b Sce U ad V are admble marce here ex x y uch ha x U ad y V he by vrue of codo 4 here ex f Ed uch ha fa x b y Hece x y C x y a b x y c C h mea ha c a b herefore c a b Codo 5 roved Oe Acce

10 KEMOKLIDZE 679 We have obaed ha he fuco : where ha form () ad he e of all admble (wh reec o ) marce afe he codo of heorem Hece he followg aeme rue heorem 3 he lace of fully vara ubgrou of he cooro hull of a drec um of oro-comlee -grou omorhc o he lace of fler of he emlace 4 Acowledgeme h udy wa uored by he gra (ASU-3/44) of Aa ereel Sae Uvery REFERENCES [] L Fuch Ife Abela Grou I Academc Pre New Yor Lodo 97 [] L Fuch Ife Abela Grou II Academc Pre New Yor Lodo 973 [3] R Baer ye of Eleme ad Characerc Subgrou of Abela Grou Proceedg Lodo Mahemacal Socey Vol -39 No h://dxdoorg//lm/-3948 [4] I Kalay Ife Abela Grou he Uvery of Mchga Pre A Arbor 969 [5] R S Lo O Fully Ivara Subgrou of Prmary Abela Grou Mchga Mahemacal Joural Vol No h://dxdoorg/37/mm/958 [6] J D Moore ad E J Hewe O Fully Ivara Subgrou of Abela -Grou Commear Mahemac Uvera Sac Paul Vol [7] R S Perce Homomorhm of Prmary Abela Grou I: oc Abela Grou (Proc Symo New Mexco Sae Uv 96) Sco Forema ad Co Chcago [8] A R Chehlov O Proecve Ivara Subgrou of Abela Grou Ve omogo Goudarveogo Uverea Maemaa Mehaa Vol 9 No ( Rua) [9] R Göbel he Characerc Subgrou of he Baer- Secer Grou Mahemache Zechrf Vol 4 No h://dxdoorg/7/bf469 [] S Ya Grho ad P A Krylov Fully vara Sub- grou Full ravy ad Homomorhm Grou of Abela Grou Algebra Joural of Mahemacal Scece (New Yor) Vol 8 No h://dxdoorg/7/ [] A Mader he Fully Ivara Subgrou of Reduced Algebracally Comac Grou Publcaoe Mahemacae Debrece Vol [] V M Myaov O Full ravy of Reduced Abela Grou I: Abela Grou ad Module No (Rua) om Sae Uvery om ( Rua) [3] A I Moaleo Cooro Hull of a Searable Grou Algebra Loga Vol 8 No ( Rua); ralao Algebra ad Logc Vol 8 No (99) [4] D K Harro Ife Abela Grou ad Homologcal Mehod Aal of Mahemac Vol 69 No h://dxdoorg/37/9788 [5] S Bazzo ad L Salce A Ideedece Reul o Cooro heore over Valuao Doma Joural of Algebra Vol 43 No 94-3 h://dxdoorg/6/abr88 [6] S Bazzo ad J Šóvče Sgma-Cooro Module over Valuao Doma Forum Mahemacum Vol No h://dxdoorg/55/forum944 [7] R Göbel S Shelah ad S L Wallu O he Lace of Cooro heore Joural of Algebra Vol 38 No 9-33 h://dxdoorg/6/abr869 [8] M Hovey Cooro Par Model Caegory Srucure ad Rereeao heory Mahemache Zechrf Vol 4 No h://dxdoorg/7/ [9] W May ad E ouba Edomorhm of Abela Grou ad he heorem of Baer ad Kalay Joural of Algebra Vol 43 No h://dxdoorg/6/-8693(76)939-3 [] Kemoldze O he Full ravy of a Cooro Hull Georga Mahemacal Joural Vol 3 No [] Kemoldze he Lace of Fully Ivara Subgrou of a Cooro Hull Georga Mahemacal Joural Vol 6 No [] Kemoldze O he Full ravy ad Fully Ivara Subgrou of Cooro Hull of Searable - Grou Joural of Mahemacal Scece (New Yor) Vol 55 No h://dxdoorg/7/ y Oe Acce