On certain classes of modules

Publicacions Ma emá iques, Vol 36 (1992), 1011-1027 . A bs ac ON CERTAIN CLASSES OF MODULES K . VARADARAJAN * Dedica ed o he me7no y o Pe e Menal Le X be any class o R-modules con aining 0 and closed unde iso no pllic images . Wi h any such X we associa e h ee classes FX, FX and ¿5X . 7 .'lie s udy o some o he closu e p ope ies o h ee classes allows Lis o ob ain cha ac e iza ion o A inian modules dualizing esul s o Cha e s . The heo y o Dual Goldie dimension as de eloped by he au ho in some o his ea lie wo k plays a c ucial ole in he p esen pape . In oduc ion Th oughou his pape all he ings R we conside will be associa i e wi h an iden i y elemen 1R ,-E 0 . Unless o he wise men ioned all he no ions such as a inianness, noe he ianness will be le sided whenwe deal wi h a ing R . The modules we conside will all be uni al le modules . In ing heo y he e a e sco es o esul s dealing wi h he s uc u e o a ing R ( esp . o a module M) assuming ce ain classes o modules (associa ed o M) posses ce ain p ope ies and ice e sa . The esul s in he p esen pape a e o a simila na u e and a e an ou come o esul s p o ed in [1], [2], [3], [4], [5] ; [6], [8] and [9] . In [1] among o he esul s A . W . Cha e s p o es he ollowing : (i) R is noe he ian i and only i e e y cyclic R-module is a di ec sum o a p ojec i e module and o a noe he ian module . (ii) Ci en an o dinal ce, i e e y cyclic R-module is a di ec sum o a p ojec i e R-module and an R-module o K ull dimension <_ a, hen he le R-module R has K ull dimension < a + 1 . *While ca ying ou his esea cli he au ho was isi ing he Ta a Ins i u e o Fun- damen al Resea ch on in i a ion om he Na ional Boa d o Highe Ma hema ics o India . Also pa o his esea ch was ca ied ou a S an o d Uni e si y whe e he au ho spen a po ion o his Sabba ical lea e . Pa ial suppo om NSERC g an A 8225 is g a e ully acknowledged . 101 2  K . VARADARAJAN In [4] P . F . Smi h, Din Van Huynh and Nguyen V . Dung gen- e alize hese esul s o Cha e s o module heo e ic se up . Le X be any class o R-modules closed unde iso no phic images and sa is ying OEX . To any such X, P . F . Smi h e all associa e h ee classes DX, HX and EX and s udy some o hei closu e p ope - ies unde sui able assump ions on X . This no only led hem o simple p oo s o he a o emen ioned esul s o Cha e s, bu also o hei module heo e ic gene aliza ions . Le N, G, K a deno e espec i ely he classes o noe he ian modules, ini ely gene a ed modules and modules o K ull dimension G a . The module heo- e ic gene aliza ions ob ained in [4] could be s a ed as ollows . (iii) G l DN = N (gene alizing (i)) . (i ) GnD & C K, :, + gene alizing (ii)) . These a e co olla ies 3 .3 and 2 .8 espec i ely in [4] . Sugges ed by "duali y" in he ca ego y R~mod o uni al le R-modules we associa e o X h ee mo e classes FX, OX and FX (see Sec ion 1 o hei de ini ion) . The s udy o some o he closu e p ope ies o hese classes leads o many in e es ing esul s "dualizing" he esul s o P . F . Smi h, Din Van Huynh and Nguyen V . Dung [4] . The objec o he p esen pape is o ca y ou he s udy o hese closu e p ope ies and p esen p oo s o he dual esul s . Fo ins an e one o he esul s we p o e using ou me hods is he ollowing : ( ) Le M be a semi-pe ec module in he sense o [13] . Assume ha ei he M is ini ely gene a ed o ha M is ini ely embedded and J(M) is small in M . Then M is a inian i and only i e e y submodule o M is a di ec sum o an injec i e moduleand an a inian module . Ac ually ) may be ega ded as wo o ms o duals o (iii) . A co olla y o ) is he ollowing cha ac e iza ion o le a inian ings . ( i) A ing R is le a inian i and only i i is semi-pe ec and e e y le ideal o R is a di ec sum o an injec i e le ideal and an a inian le ideal . 1 . The classes FX, AX and I'X We will be wo king in he ca ego y R-mod o uni a y le R-modules . The classes X o R-modules we conside will always be assumed o sa is y he ollowing condi ions a and b . a . ME_X . M' - M = :> NI'cX . b . OEX . P oo . (i) S aigh o wa d . Oi CERTAINCLASSES oH MODULES  1013 To any such X, P . F . Smi h e all [4] associa ed h ee clases o modules ( hough hey wo ked in he ca ego y mod- .R, o igh R-modules) . Be o e ecalling he de ini ion o h ee classes, we i s explain he no a ion ha we will be adop ing . Fo any A/IcR,- nod, we w i e N < 11NI o indica e ha N is a submodule o 111 ; Né1VI o indica e ha N is an essen ial submodule o 111 - and N « AI o deno e ha N is a . small submodule o 1VI . The h ee classes DX, TIX and EX we e de ined as ollows in [4] . DX = {AIcR-modIN < M =111=K®L wi h N<K and K/N~!} . HX = {AIcR-modIN < M =~> II/NcX } EX = {AIcR-modIN :Al = :> AI/NEX} . Sugges ed by "duali y" we in oduce he ollowing clases : FX= {AIcR-modIN < M  M=K®L wi h K<N and N/KcX} . FX = {AIcR-modiN < AI  NcX} OX = {McR-modIN « M  NcX} . As in [4] when he ing R is clea om he con ex , M, Z, P, I, C, G, N, A, U ; K will deno e espec i ely he classes o all R-modules, he ze o modules, p ojec i e modules, injec i e modules, semi-simple mod- ules, ini ely gene a ed modules, noe he ian modules, a inian modules, modules o ini e uni o m di nension and modules wi h K ull di nension <_ a . Recall [11] ha 116R-mod is said o be o dual Goldie di nension <_ k i h ee exis s no su jec i e map M _ W > N l x ... xN, ., wi h each Ni :~ 0 and >_ (k + 1) . He e k is an in ege >_ 0 . The class o modules o dual Goldie di nension <_ k will be deno ed by H k . We w i e S o he class cons i u ed by he simple modules oge he wi h he ze o mod- ule . We will nos ly be ollowing he no a ion and e minology in [4] . The class o modules o ini e dual Goldie di nension (o co ank) will be deno ed by H . Lemma 1 .1 . Le X, Y be classes o R-modules (i) I X C_ Y hen LX C_ Y whe e L s ands o any one o he symbols D, H, E, , F o 0 . (ii) FX = F(FX) C X . (iii) C C FX . (i ) Fx C F(I (D X) C F(I ® X) = F(X) = (X) C A(x) . ( ) i n x c F((D x) . 101 4  K . VARADARAJAN (ii) F om he- e y de ini ion o FX i is clea ha FX C X . Hence (i) abo e yields F(FX) C FX . Le McFX and N <_ M . Le N' < N . Then N' < M ; hence N'6X yielding NcFX . This in u n implies ha AJEF(FX) ; hence FX C F(FX) . (iii) Le McC and N < M . Then M = N ® L o some L< M . Hence he choice K = N ul ills he equi emen o M o be in FX . (i ) Since X C_ I ® X, om (i) we ge FX C F(I ® X) . Le McF(I ® X) and N <_ M .  Since McF(I ® X)we ge NeI ® X . Thus M = 0 ® M and N/0 - NeI ®X . This means McF(I ® X) . Hence F(I ® X) C_ F(I ® X) . Because o (i), o p o e he equali y F(I ® X) =FX we ha e only o show ha F(I ®_ X) C FX . Le Mc (I ®X) and N < M . Then M = K ® L wi h K <_ N and N/K6I ® X . F om K < N we ge N = K ® (L nN) ; hence L n N -N%K eI ®X . This yields L n N =A®B wi h AeI, BcX . Since AeI and A _< L we could w i e L = A ®C wi h CeM . Thus M = K ®L = K ®A®C . Also K®A<N . HenceN=K®A® (CnN) . AlsoA_<LnN==> L n N = A ®(C n N n L) = A ®(C n N) since C < L . F om A®B=LnN=A®(CnN) wege B-(LnN)/A-CnN yielding C n NeX . Also M = K ®A ® C wi h K ®A <_ N and N/(K ® A) -C n NeX . This p o es ha Mcl'X . Hence F(I ® X)C FX . To comple e he p oo o i ) we ha e only o show ha X C AX . Le M6F_X and N « M . Then M = K ® L wi h K < N and N/KEX . F om K <_ N«M we ge K K M . Since K is a di ec summand o M his implies ha K = 0 ; hence NeX showing ha Mc0_X . ( ) Le MeI n FX and N _< M . F om MEFX we ge M = K ® L wi h K <_ N and N/KEX . Then N = K ® (L n N) yielding N/K - Ln NEX . Also MeI ==> KeI ; hence NEI ® X . This means M6F(I ® X) yielding I n FXC F(I ® X) . s Be o e s a ing u he esul s le us ecall om [41 he de ini ion o SX , QX and PX . SX = {NIN <M, McX} . QX = {M/NIN < M, M6X} . PX = {MI he e exis s a ini e chain 0 = No < N l <  <Nk = M wi h Ni/N2_leX o 1 < i < k}- X is said o be S ( esp Q o P) closed i SX C_ X ( esp . QX C X o PX C X) . ON CERTAINCLASSES OF MODULES  1015 Lemma 1 .2 . Le X be a class o R-modules . Then (i) FX, ~X, FX a e all S-closed . (ii) I X is S-closed, hen X C_ FX and XC C_ OX . (iii) FX ® X = FX i X is {S, P}-closed . (i ) F(I ® X) = (I ® X)n FX i X is {S, P}-closed . ( ) FX is Q-closed i X is Q-closed . P oo .. (i) Tha F_X is S-closed is clea . Le MEIX and M' < M . Le N'« M' . Then N' «M and hence N'EX . This means M'EOX . Le MEFX and M' <_ Al . Le N <_ M' . F om MEFX we ge M=K®Lwi hK_<NandN/KEX . F om K_<N<M'we ge M'= K ® (M' n L) . Clea ly N/KEX ; hence M'cI'X . (ii) Le MEX and N <_ M . Since X is S-closed we ha e NEX . Thus M = 0 ® M wi h N/0 -NEX, yielding MEFX . Hence X C FX . Le MEX _C .  Then he e exis s a K _< M wi h KE_X and M/KEC . Le N « M . Then N <_ J(M), he Jacobson adi- cal o M . I l : M --> M/K deno es he canonical quo ien map we ge n(N) < 97(J(M)) < J(M/K) = 0 Since MlKEC . Hence N _< K . Since X is S-closed we ge NeX . Thus McAX yielding _XC C_ 21X . (iii) Le MEFX ® X, say M = A® B wi h AEI'X, BEX . Le N <_ Al . Since AEFX we ge A= K®L wi h K < NnA and (NnA)/KeX . Thus M = K (D L ®B and M/A - BeX . The exac ness o 0 -, N/(N n A) -> M/A oge he wi h he S-closed na u e o X yields N/(N nA)cX . The exac ness o 0 -> (N n A)/K N/K -> N/ (NnA) -> 0 and he P-closed na u e o X imply ha N/KEX . Hence MEFX, yielding I'X ® X C FX . The e e se inclusion FX C_ FX® _X is ob ious . (i ) F om lemma 1 .1(ii) and (i ) we see ha F(1E)X) C (I® X)n X . We can w i e M = A ® B wi h AEI, BeX . F om lemma 1 .2(i) we see ha AEFX . Le N _< M . Since AEFX we ge A = K ® L wi h K _< A n N and An N/KEX . Hence M = K ® L ®B . F om K <_ N we ge N = K ® (L ® B) n N . Also AEI => KeI . The exac ness o 0 -> N/(N n A) -> M/A and 0 -> (A n N)/K -~ N/K -> N/ (A n N) -~ 0 and {S, P}-closedness o X immedia ely yield N/KEX . Bu N/K-Nn(L®B) . Hence NEI®X, p o ing ha MEF(I ® X) . Hence (I (D X)n FX C_ F(I ® X) . ( ) Le MEFX and N < M . Any submodule o M/N is o he o m L/N wi h N _< L _< M . F om MEFX we in e LEX . Since X is Q-closed we ge LINEX . This implies ha M/NEFX . Rema ks 1 .3 . Lemma 1 .1( ) in [4] also asse s ha EX is S-closed 101 6  K . VARADARAJAN i _X is S-closed .  The dual esul i i we e ue would, be , ha AX is Q-closed whene e X is Q closed . We now . gi e an easy esample o show ha he dual esul is no ue . Le Z deno e he class consis ing o he ze o modules in Z-mod . Clea ly Z is Q-closed . Also AZ = {M6Z-mod jJ(M) = 0} . Clea ly Z6OZ, bu Z p 2 1 OZ o any p ime p . This shows ha áZ is no Q-closed . P oposi ion 1 .4 . Le X be any {S, P}-closed amily o modules . Then FX =FX® X ® (P nFX) . P oo . We need only p o e he inclusion FX ® X ® (P nFX)C FX . F om lemma 1 .2(i ) we ha e 1'X ® X = FX . Hence i su ices o p o e ha FX ® (P n FX) C_ FX . Le M = A®B wi h Ac X and BeP n FX . Le N <_ M and PB M =A® B --> B he p ojec ion on o B . F om Be X we ge B . =B l ® B2 wi h Bl < PB(N) and PB(N)/BlEX . F om BcP we ge B 1 E P and B2EP . Le n =PBINn(A®B1) : Nn(A®B 1 ) -~ B, .  Since B l <PB (N) we see ha a : N n (A ®BO -> B lis on o . Since B 1 6P, he e exis s a spli ing s : B l -> N n (A ® Bl) o a . Le N' = s(Bl) . Then N n (A ® Bl) = N I ®'Ke a= N' - ® (Nn A) . F om AcFX we' ge A = A 1 ® A2 wi h A,'<_ N n A and (N n A)/A l eX . Again, NnA = Al ®(NnAnA 2 ) = A1®(NnA2) yields NnA 2 - (NnA)/AleX . Conside , pB/A ®B l : A® Bl  + B, . Clea ly s is also a spli ing o pB/A ® B, . Since Ke pB/A ®B l =A we see ha A® N' is - ano he in e nal di ec sum ep esen a ion o A® B, . Hence M = A®B= A®Bl®B2=A®N'®B2=A1® A2 .®N'® B 2 . Since Al ®N'<N wegé N=A l ®N'eNn(A2TB2) . .Le y=PBINn(A2®B2) : N n (A2 ® B2) - B2 . Since pB(Al ® N') < B l and B= B l ® B2 we see ha PB(N) nB2 = PB((A2 ® B2) n N) = Image y . Bu PB(N) _ B l ® (PB(N) n B2) ; hence Image y = pB(N) n B2 - PB(N)/B, is in X . Aslo Ke y = N n A 2 EX . Since X is P-closed we ge N n(A2 ® B2)6X . Also N%(A1 ® N') - N n(A 2 (D'B2)cX . This shows ha MEFX . Thus FX® X ® (P n FX) CFX . This comple es he p oó o p oposi ion 1 .4 . E Lemma 1 .5 . I X is Q-closed hen OX is closed u ide minimal epi- mo phic images . P oo . Le McAX and M -~ M" aminimal epimo phism (Le . Ke e « M) . Then N" « M"  c -1 (N") « M . In pa icula N" « M" =~> e -1 (N") « M => e -1 (N")cX  N"eX (since X is Q-closed) . This p o es ha M"6AX . Be o e p oceeding u he we need o ecall some de ini ions and e- sul s om [7], [111,J12] ; [13] . Le N < M . Then K < M is called a supplemen o N in M i ON CEI TAIN CLASSES O MODULES  1017 (a) K+N=hand (b) K'<K,K'+N=A,1=~> K'=K . I is known ha K is a supplemen o N in M i and only i K+N = M and K 1N « K (Le nma 6 .2 in [13]) . In [13] we called a module M semi- pe ec i o e e y N < AJ he e exis s a supplemen in M (De ini ion 6 .6 in [13]) . In [11] we e e ed o his as p ope y (P l ) o M . The module M is said o ha e p ope y (P2) i o any L _< Al1, N <_ M sa is ying L + N = M he e exis s a supplemen K o N in AJ sa is ying K _< L . I M has p ope y (Pi) hen any quo ien module o M has p ope y (P i ) o i = l, 2 (P oposi ion 6 .20 in [13] and P oposi ion 2 .29 in [11]) . Clea ly P 2 => PI . Lemma 1.6 . Le X be Q-closed and AJEAX . Assume u he ha AJ has p ope y (Pi) . Then e e y epimo phic image o M is in OX . P oo .. Le q : M -> AJ" be any epi no phism and N = Ke 91 . Le K be a supplemen o N in M . Then K + N = AJ and K 1 N« K . In pa icula 71/K : K -> M" is a minimal epi no phism . F om lemma 1 .2(i) we ge KE,~,X . Now lemma 1.5 yields M"eáX . Example 1 .7 . (a) Le T deno e he class o o sion abelian g oups . In Z-mod, T is {S, P, Q}-closed . In [4] he class DT is comple ely de e mined (P oposi ion 1 .6 o [4]) . I is easy e see ha ET = A%1 and ha HT =T= FT . Fo any AJcZ-mod le J(A4) deno e i s Jacobson adical . Since J(A11) is he sum o all s nall submodules o All we see inmmedia ely ha AT= {MEZ -mod jJ(A11)cT} . F om lemma 1 .2(i) we know ha FT is S-closed . Since he only di ec su nmands o Z a e 0 and Z i ollows ha Z 1 F T . Combining his wi h he S-closed na u e o F T we see ha FT C_ T . Also lem na 1 .2(ii) implies T C FT . Hence FT =T . (b) Le T' deno e he class o o sion ee abelian g oups . Then T' is S-closed . I is i ial o see ha FT' = T' . Suppose A FT' . Since he only o sion ee ac o g oup o a o sion abelian g oup is 0 we see ha any N < (AJ) is a di ec summand o 1Vl (he e (M) deno es i e o sion subg oup o M) . I ollows ha any N <_ (M) is a di ec summand o (M) and ha (M) i sel is a di ec summand o M . Thus (M)cC and AJ = (M) ® L wi h LcT' . This yields FT' C_ C® T' . Also AcC <~--> A = (A) and p (A) is a ec o space o e Z p o e e y p ime p . Le M = A® B wi h AcC and BET' . Le N <AL Then (N) < (1V1) = A . Since AEC we ge A = (N) ® L and 101 8  K . VARADARAJAN bo h (N) and L will be in C . F om M = A® B= (N) ® L ®B and N/ (N)cT_' we see ha McI'T' . Hence C ®T' C_ I'T' . Using he e e se inclusion al eady p o ed we ge FT' = CT T' . F om lemma 1 .1(i ) we ha e FT' C OT' . We will ac ually gi e a com- ple e cha ac e iza ion o he class áT' o n which i will ollow i n nedi- a ely ha he inclusion FT' C_ áT' is a s ic inclusion . Le M6AT ' . Suppose o some p ime p, he p-p ima y o sion p (M) o M is non-ze o . Then he e exis s a copy o Z p in p (M) . Suppose N <_ M sa is ies Z p + N = M . Ei he N 1 Z p = Z p o N n Z p = 0, in he o me case N = M and in he la e case M = N ® Z p . I o all N <_ M sa is ying Z p + N = M we ha e N = M, hen Z p « M and his con adices he assump ion ha McAT' . Hence M = Z p ® N o some N <_ M . Thuswe ha e shown ha i p (M) :7É 0, any copy o Z p in p (M) is a di ec summand o M . In pa icula his implies ha he e a e no elemen s o o de p 2 in p (M), hence p (M) is a ec o space o e Z p . Hence (M) =® p p (M) is in C . We claim ha (4)  n T' = {MeZ-mod / any Z p < M o any p ime p is a di ec summand o M} . Because o he obse a ions in he ea lie pa ag aph, o p o e (4) we ha e only o show ha i MeZ-mod has he p ope y men ioned in he igh hand side o (4) and i N « M hen NcT' . I on he con a y he e is an N « M wi h N 0 T', hen p (N) ,-É 0 o some p ime p . Then he e is a copy o Z p in p (N) . Since N « M i will ollow ha his copy o Z p is small in M . Howe e , any Z p < M being a di ec summand o M canno be small in M . F om (4) we see ha (di ec p oduc o e all p imos) is in AT' . Howe e , (M) =® p Z p and i is well-known ha (M) does no spli o o n M . Hence 111 « FT' . This p o es ha he inclusion FT' C áT' is s ic . 2 . S udy o AX when X = A n H, Fo esul s on dual Goldie dimension o co ank he eade may e e o [7], [111 . As al eady ema ked in [11], i he dual Goldie dimension ON CER .TAIN CLASSES OF MODULES  101 9 o M is in ini o we canno asse ha he e exis s a su jéc i e map cp M -> 11' 1 Ni wi h each Ni :,A 0 . (See P oposi ion 1.6 in [11]) . All we can asse in his case is ha , gi en any in ege d >_ 1 we can ind a ce ain su jec ion 9 : M --> Il,q- 1 Lj wi h each Lj 7~ 0 ( he modules L .- in gene al will depend on d ) . This di e en beha iou o dual Goldie dimension as compa ed o Goldie dimension necessi a es many changos in he o mula ion and in he p oo s o esul s dual o hose ob ained in Sec ion 2 o [4] whe e he heo y o Goldie dimension plays a c ucial ole . We i s obse e ha he class H kis Q-closed . Lemma 2 .1 . Le X be Q-closed wi h X C_ H k . Le AllcOX and N _< M sa is y N + J(A11) = AJ . Assione ha Al has p ope y (P1) . Then M/NcH k . P oo ., Le l : Al -> M/N deno e he quo ien map . F om N + J(M) = M we ge l(J(M)) = M/N . Hence J(M/N)= M/N . Suppose i possible ha M/N has dual Goldie dimension > k . Then he e exis s a su jec ion cp : M/N ---> A1 x . . . x Ae wi h 2 > k and each A j :y~ 0 . F om J(M/N) = M/N we ge J(A ;) = Aj o 1<_ j <_ Q . Since J(Aj) = Aj ,-á 0 and J(A ;) is he sum o all small submodules o A j we see ha he e exis s a Bj « Aj wi h B . :~ 0 . Then B 1 x . . . x Bg « A 1 x . . . x Ae . F om lemma 1 .6 we ge A1 x . . . x A e cOX . This implies B 1 x . . . x BQcX . This con adic s he assump ion ha X C_ H k , since co ank B 1 x . . . x BQ > 2 > k . Co olla y 2 .2 . Suppose X is Q-closed and _X C_ H k . Le McáX . Suppose M Izas p ope y (P1) and, sa i .s ies,I(M) = M . Then McH k . P oo :: Choose N = 0 in lemma 2 .1 . P oposi ion 2 .3 . Le X be Q-closed wi h X C_ H k . Le Mci1X and assume ha M has p ope y (P1) . Then he e exis s an N6X such ha 114'/N = B e H wi h B¿C and HEH k n OX . P oo .: Le L be a supplemen o . .I(A4) in AJ . Then L+ J(M) = M and LnJ(M) « L . Also L/(L n J(M)) - AJ/J(AJ)e0X by lemlna 1 .6 . Since OX is S-closed (lemma 1 .2(i)) we ge LcOX . F om LnJ(M) « L we ge Ln J(M)cX . Since M/J(M) has p ope y (P1) (P oposi ion 6 .1 in [13]) and J(M/J(M)) = 0 om p oposi ion 3 .3 in [11] we see ha M/J(M)cC . Hence L/(LnJ(M))EC . F om lemma 2 .1 we ge MILc_H k . l we se N = LnJ(M) we ge NeX and 0 -> L/N -> M/N -> MIL -> 0 exac . 102 6  K . VARADARAJAN We will now cha ac e ize he class V(T') . We will show ha V(! :') = {McTI J(M) = 0} . Le McV(T') . We will show ha 0 is he only small submodule o M . Then i ollows ha J(M) = 0 . Suppose on he con a y 0 :,A N « M . Since V(T') C_ T' we ha e MeT' . Hence NcT' . This means he e is a copy o Z in N . Conside he subg oup 2Z o Z . F om2Z <_ Z <_ N « 111 we ge 2Z « M . Now, M/2Z has non-ze o 2 o sion, con adic ing he ac ha McV(T') . Con e sely any MeT' wi h J(M) = 0 is clea ly in V(T') because hen 0 is he only small submodule o M and M/0 - MeT' . This p o es (5) . F om (5) we see ha he inclusion V(! :') C T' is a s ic inclusion ; because Q6T' bu Q « V(T') since J(Q) = Q . We included in o ma ion en he classes LT, VT, LT' and VT' o comple e he examples discussed in 1 .7 . Re e ences 1 .  A . W . CHATTERS ; A cha ac e iza ion o igh Noe he ian ings, Qua e ly J . Ma h . Ox o d 33 (1982), 65-69 . 2 .  DINH VAN HUYNH AND PHAN DAN, On ings wi h es ic ed min- imum condi ion, A ch . Ma h . 51 (1988), 313-326 . 3 .  DINH VAN HUYNH ; NGUYEN V . DUNG AND PATRICK F . SMITH, Rings cha ac e ized by he igh ideals o cyclic modules, P oceed- ings o he Edinbu gh Ma hema ical Socie y 32 (1989), 355-362 . 4 .  PATRICK F . SMITH, DIN VAN HUYNH AND NGUYEN V . DUNG, A cha ac e iza ion o Noe he ian modules, Qua . J . Ma h . Ox o d 41 (1990),225-235 . 5 .  DIN VAN HUYNH, NGUYEN V . DUNG, AND PATRICK F . SMITH, A cha ac e iza ion o ings wi h K ull dimension, J . Alg . 132 (1990), 104-112 . 6 .  DINH VAN HUYNH AND NGUYEN V . DUNG, A cha ac e iza ion o a inian ings, Clasgow Ma h . J . 30 (1988), 67-73 . 7 . B . SARATH AND K . VARADARAJAN, Dual Goldie dimension II, Communica ions in Alg . 7 (1979), 1885-1899 . 8 .  P . F . SMITH, Some ings which a e cha ac e ized by ini ely gene - a ed modules ; Qua . J . Ma h . Ox o d 29 (1978), 101-109 . 9 .  P . F . SMITH, Rings cha ac e ized by he cyclic modules, Canadian J . Ma h . 30 (1978), 98-111 . 10 .  P . VAMOS, The dual o he no ion o ini ely gene a ed, J . London Ma h . Soc . 43 (1969) . ON CERTAIN CLASSES OF MODULES  1027 11 . K . VARADARAJAN, Dual Goldie dilnension, Communica ions in Alg . 7 (1979), 565-610 . 12 . K . VARADARAJAN, Modules wi h supplemen s, Pac . J . o Ma h . 82 (1979),559-564 . 13 . K . VARADARAJAN AND P . R . WANI, Modules o e endomo phism ings 11, Ac a llla h . HungaHca 53 (1989), 309-337 . 14 . K . VARADARAJAN, Hop ian and co-Hop ian objec s ( o appea ) . Depa men o Ma hema ics and S a is ics The Uni e si y o Calga y 2500 Uni e si y D i e N .W . Calga y, Albe a LANADA T2N 1N4 P ime a e sió ebuda el 15 de No emb e de 1991, da e a e sió ebuda el 2de Ma 4 de 1992