On certain classes of modules

Publicacions Matemátiques, Vol 36 (1992), 1011-1027 . A bstract ON CERTAIN CLASSES OF MODULES K . VARADARAJAN * Dedicated to the me7nory of Pere Menal Let X be any class of R-modules containing 0 and closed under isornorpllic images . With any such X we associate three classes FX, FX and ¿5X . 7 .'lie study of some of the closure properties of three classes allows Lis to obtain characterization of Artinian modules dualizing results of Chatters . The theory of Dual Goldie dimension as developed by the author in some of his earlier work plays a crucial role in the present paper . Introduction Throughout this paper all the rings R we consider will be associative with an identity element 1R ,-E 0 . Unless otherwise mentioned all the notions such as artinianness, noetherianness will be left sided whenwe deal with a ring R . The modules we consider will all be unital left modules . In ring theory there are scores of results dealing with the structure of a ring R (resp . of a module M) assuming certain classes of modules (associated to M) posses certain properties and viceversa . The results in the present paper are of a similar nature and are an outcome of results proved in [1], [2], [3], [4], [5] ; [6], [8] and [9] . In [1] among other results A . W . Chatters proves the following : (i) R is noetherian if and only if every cyclic R-module is a direct sum of a projective module and of a noetherian module . (ii) Civen an ordinal ce, if every cyclic R-module is a direct sum of a projective R-module and an R-module of Krull dimension <_ a, then the left R-module R has Krull dimension < a + 1 . *While carrying out this researcli the author was visiting the Tata Institute of Fundamental Research on invitation from the National Board for Higher Mathematics of India . Also part of this research was carried out at Stanford University where the author spent a portion of his Sabbatical leave . Partial support from NSERC grant A 8225 is gratefully acknowledged . 101 2  K . VARADARAJAN In [4] P . F . Smith, Din Van Huynh and Nguyen V . Dung generalize these results of Chatters to module theoretic set up . Let X be any class of R-modules closed under isornorphic images and satisfying OEX . To any such X, P . F . Smith et all associate three classes DX, HX and EX and study some of their closure properties under suitable assumptions on X . This not only led them to simpler proofs of the aforementioned results of Chatters, but also to their module theoretic generalizations . Let N, G, K a denote respectively the classes of noetherian modules, finitely generated modules and modules of Krull dimension G a . The module theoretic generalizations obtained in [4] could be stated as follows . (iii) G fl DN = N (generalizing (i)) . (iv) GnD & C K, :, +r generalizing (ii)) . These are corollaries 3 .3 and 2 .8 respectively in [4] . Suggested by "duality" in the category R~mod of unital left R-modules we associate to X three more classes FX, OX and FX (see Section 1 for their definition) . The study of some of the closure properties of these classes leads to many interesting results "dualizing" the results of P . F . Smith, Din Van Huynh and Nguyen V . Dung [4] . The object of the present paper is to carry out the study of these closure properties and present proofs of the dual results . For instante one of the results we prove using our methods is the following : (v) Let M be a semi-perfect module in the sense of [13] . Assume that either M is finitely generated or that M is finitely embedded and J(M) is small in M . Then M is artinian if and only if every submodule of M is a direct sum of an injective moduleand an artinian module . Actually v) may be regarded as two forms of duals of (iii) . A corollary of v) is the following characterization of left artinian rings . (vi) A ring R is left artinian if and only if it is semi-perfect and every left ideal of R is a direct sum of an injective left ideal and an artinian left ideal . 1 . The classes FX, AX and I'X We will be working in the category R-mod of unitary left R-modules . The classes X of R-modules we consider will always be assumed to satisfy the following conditions a and b . a . ME_X . M' - M = :> NI'cX . b . OEX . Proof . (i) Straight forward . Oiv CERTAINCLASSES oH MODULES  1013 To any such X, P . F . Smith et all [4] associated three clases of modules (though they worked in the category mod- .R, of right R-modules) . Before recalling the definition of three classes, we first explain the notation that we will be adopting . For any A/IcR,-rnod, we write N < 11NI to indicate that N is a submodule of 111 ; Né1VI to indicate that N is an essential submodule of 111 - and N « AI to denote that N is a . small submodule of 1VI . The three classes DX, TIX and EX were defined as follows in [4] . DX = {AIcR-modIN < M =111=K®L with N<K and K/N~!} . HX = {AIcR-modIN < M =~> II/NcX } EX = {AIcR-modIN :Al = :> AI/NEX} . Suggested by "duality" we introduce the following clases : FX= {AIcR-modIN < M  M=K®L with K<N and N/KcX} . FX = {AIcR-modiN < AI  NcX} OX = {McR-modIN « M  NcX} . As in [4] when the ring R is clear from the context, M, Z, P, I, C, G, N, A, U ; K tt will denote respectively the classes of all R-modules, the zero modules, projective modules, injective modules, semi-simple modules, finitely generated modules, noetherian modules, artinian modules, modules of finite uniform dirnension and modules with Krull dirnension <_ a . Recall [11] that 116R-mod is said to be of dual Goldie dirnension <_ k if three exists no surjective map M _ W > N l x ... xN, ., with each Ni :~ 0 and r >_ (k + 1) . Here k is an integer >_ 0 . The class of modules of dual Goldie dirnension <_ k will be denoted by H k . We write S for the class constituted by the simple modules together with the zero module . We will rnostly be following the notation and terminology in [4] . The class of modules of finite dual Goldie dirnension (or corank) will be denoted by H . Lemma 1 .1 . Let X, Y be classes of R-modules (i) If X C_ Y then LX C_ Y where L stands for any one of the symbols D, H, E, r, F or 0 . (ii) FX = F(FX) C X . (iii) C C FX . (iv) Fx C F(I (D X) C F(I ® X) = F(X) = r(X) C A(x) . (v) i n rx c F((D x) . 101 4  K . VARADARAJAN (ii) From the-very definition of FX it is clear that FX C X . Hence (i) above yields F(FX) C FX . Let McFX and N <_ M . Let N' < N . Then N' < M ; hence N'6X yielding NcFX . This in turn implies that AJEF(FX) ; hence FX C F(FX) . (iii) Let McC and N < M . Then M = N ® L for some L< M . Hence the choice K = N fulfills the requirement for M to be in FX . (iv) Since X C_ I ® X, from (i) we get FX C F(I ® X) . Let McF(I ® X) and N <_ M .  Since McF(I ® X)we get 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 of (i), to prove the equality F(I ® X) =FX we have only to show that F(I ®_ X) C FX . Let Mcr(I ®X) and N < M . Then M = K ® L with K <_ N and N/K6I ® X . From K < N we get N = K ® (L nN) ; hence L n N -N%K eI ®X . This yields L n N =A®B with AeI, BcX . Since AeI and A _< L we could write L = A ®C with 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 . From A®B=LnN=A®(CnN) wegetB-(LnN)/A-CnN yielding C n NeX . Also M = K ®A ® C with K ®A <_ N and N/(K ® A) -C n NeX . This proves that Mcl'X . Hence F(I ® X)C FX . To complete the proof of iv) we have only to show that rX C AX . Let M6F_X and N « M . Then M = K ® L with K < N and N/KEX . From K <_ N«M we get K K M . Since K is a direct summand of M this implies that K = 0 ; hence NeX showing that Mc0_X . (v) Let MeI n FX and N _< M . From MEFX we get M = K ® L with 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 Before stating further results let us recall from [41 the definition of SX , QX and PX . SX = {NIN <M, McX} . QX = {M/NIN < M, M6X} . PX = {MI there exists a finite chain 0 = No < N l <  <Nk = M with Ni/N2_leX for 1 < i < k}- X is said to be S (resp Q or P) closed if SX C_ X (resp . QX C X or PX C X) . ON CERTAINCLASSES OF MODULES  1015 Lemma 1 .2 . Let X be a class of R-modules . Then (i) FX, ~X, FX are all S-closed . (ii) IfX is S-closed, then X C_ FX and XC C_ OX . (iii) FX ® X = FX if X is {S, P}-closed . (iv) F(I ® X) = (I ® X)n FX if X is {S, P}-closed . (v) FX is Q-closed if X is Q-closed . Proof .. (i) That F_X is S-closed is clear . Let MEIX and M' < M . Let N'« M' . Then N' «M and hence N'EX . This means M'EOX . Let MEFX and M' <_ Al . Let N <_ M' . From MEFX we get M=K®LwithK_<NandN/KEX . From K_<N<M'we get M'= K ® (M' n L) . Clearly N/KEX ; hence M'cI'X . (ii) Let MEX and N <_ M . Since X is S-closed we have NEX . Thus M = 0 ® M with N/0 -NEX, yielding MEFX . Hence X C FX . Let MEX _C .  Then there exists a K _< M with KE_X and M/KEC . Let N « M . Then N <_ J(M), the Jacobson radical of M . If rl : M --> M/K denotes the canonical quotient map we get n(N) < 97(J(M)) < J(M/K) = 0 Since MlKEC . Hence N _< K . Since X is S-closed we get NeX . Thus McAX yielding _XC C_ 21X . (iii) Let MEFX ® X, say M = A® B with AEI'X, BEX . Let N <_ Al . Since AEFX we get A= K®L with K < NnA and (NnA)/KeX . Thus M = K (D L ®B and M/A - BeX . The exactness of 0 -, N/(N n A) -> M/A together with the S-closed nature of X yields N/(N nA)cX . The exactness of 0 -> (N n A)/K N/K -> N/ (NnA) -> 0 and the P-closed nature of X imply that N/KEX . Hence MEFX, yielding I'X ® X C FX . The reverse inclusion FX C_ FX® _X is obvious . (iv) From lemma 1 .1(ii) and (iv) we see that F(1E)X) C (I® X)nrX . We can write M = A ® B with AEI, BeX . From lemma 1 .2(i) we see that AEFX . Let N _< M . Since AEFX we get A = K ® L with K _< A n N and An N/KEX . Hence M = K ® L ®B . From K <_ N we get N = K ® (L ® B) n N . Also AEI => KeI . The exactness of 0 -> N/(N n A) -> M/A and 0 -> (A n N)/K -~ N/K -> N/ (A n N) -~ 0 and {S, P}-closedness of X immediately yield N/KEX . But N/K-Nn(L®B) . Hence NEI®X, proving that MEF(I ® X) . Hence (I (D X)n FX C_ F(I ® X) . (v) Let MEFX and N < M . Any submodule of M/N is of the form L/N with N _< L _< M . From MEFX we infer LEX . Since X is Q-closed we get LINEX . This implies that M/NEFX . Remarks 1 .3 . Lemma 1 .1(v) in [4] also asserts that EX is S-closed 101 6  K . VARADARAJAN if _X is S-closed .  The dual result if it were true would, be , that AX is Q-closed whenever X is Q closed . We now . give an easy esample to show that the dual result is not true . Let Z denote the class consisting of the zero modules in Z-mod . Clearly Z is Q-closed . Also AZ = {M6Z-mod jJ(M) = 0} . Clearly Z6OZ, but Z p 2 1 OZ for any prime p . This shows that áZ is not Q-closed . Proposition 1 .4 . Let X be any {S, P}-closed family of modules . Then FX =FX® X ® (P nFX) . Proof . We need only prove the inclusion FX ® X ® (P nFX)C FX . From lemma 1 .2(iv) we have 1'X ® X = FX . Hence it suffices to prove that FX ® (P n FX) C_ FX . Let M = A®B with AcrX and BeP n FX . Let N <_ M and PB M =A® B --> B the projection onto B . From BerX we get B . =B l ® B2 with Bl < PB(N) and PB(N)/BlEX . From BcP we get B 1 E P and B2EP . Let n =PBINn(A®B1) : Nn(A®B 1 ) -~ B, .  Since B l <PB (N) we see that a : N n (A ®BO -> B lis onto . Since B 1 6P, there exists a splitting s : B l -> N n (A ® Bl) of a . Let N' = s(Bl) . Then N n (A ® Bl) = N I ®'Ker a= N' - ® (Nn A) . From AcFX we' get A = A 1 ® A2 with A,'<_ N n A and (N n A)/A l eX . Again, NnA = Al ®(NnAnA 2 ) = A1®(NnA2) yields NnA 2 - (NnA)/AleX . Consider, pB/A ®B l : A® Bl  + B, . Clearly s is also a splitting for pB/A ® B, . Since Ker pB/A ®B l =A we see that A® N' is - another internal direct sum representation for A® B, . Hence M = A®B= A®Bl®B2=A®N'®B2=A1® A2 .®N'® B 2 . Since Al ®N'<N wegétN=A l ®N'eNn(A2TB2) . .Lety=PBINn(A2®B2) : N n (A2 ® B2) - B2 . Since pB(Al ® N') < B l and B= B l ® B2 we see that PB(N) nB2 = PB((A2 ® B2) n N) = Image y . But PB(N) _ B l ® (PB(N) n B2) ; hence Image y = pB(N) n B2 - PB(N)/B, is in X . Aslo Ker y = N n A 2 EX . Since X is P-closed we get N n(A2 ® B2)6X . Also N%(A1 ® N') - N n(A 2 (D'B2)cX . This shows that MEFX . Thus FX® X ® (P n FX) CFX . This completes the proóf of proposition 1 .4 . E Lemma 1 .5 . If X is Q-closed then OX is closed urider minimal epimorphic images . Proof . Let McAX and M -~ M" aminimal epimorphism (Le . Ker e « M) . Then N" « M"  c -1 (N") « M . In particular N" « M" =~> e -1 (N") « M => e -1 (N")cX  N"eX (since X is Q-closed) . This proves that M"6AX . Before proceeding further we need to recall some definitions and results from [7], [111,J12] ; [13] . Let N < M . Then K < M is called a supplement of N in M if ON CEItTAIN CLASSES Or MODULES  1017 (a) K+N=hand (b) K'<K,K'+N=A,1=~> K'=K . It is known that K is a supplement of N in M if and only if K+N = M and Kf1N « K (Lernma 6 .2 in [13]) . In [13] we called a module M semiperfect if for every N < AJ there exists a supplement in M (Definition 6 .6 in [13]) . In [11] we referred to this as property (P l ) for M . The module M is said to have property (P2) if for any L _< Al1, N <_ M satisfying L + N = M there exists a supplement K of N in AJ satisfying K _< L . If M has property (Pi) then any quotient module of M has property (P i ) for i = l, 2 (Proposition 6 .20 in [13] and Proposition 2 .29 in [11]) . Clearly P 2 => PI . Lemma 1.6 . Let X be Q-closed and AJEAX . Assume further that AJ has property (Pi) . Then every epimorphic image of M is in OX . Proof .. Let q : M -> AJ" be any epirnorphism and N = Ker 91 . Let K be a supplement of N in M . Then K + N = AJ and K f1 N« K . In particular 71/K : K -> M" is a minimal epirnorphism . From lemma 1 .2(i) we get KE,~,X . Now lemma 1.5 yields M"eáX . Example 1 .7 . (a) Let T denote the class of torsion abelian groups . In Z-mod, T is {S, P, Q}-closed . In [4] the class DT is completely determined (Proposition 1 .6 of [4]) . It is easy te see that ET = A%1 and that HT =T= FT . For any AJcZ-mod let J(A4) denote its Jacobson radical . Since J(A11) is the sum of all srnall submodules of All we see inmmediately that AT= {MEZ -mod jJ(A11)cT} . From lemma 1 .2(i) we know that FT is S-closed . Since the only direct surnmands of Z are 0 and Z it follows that Z 1 F T . Combining this with the S-closed nature of F T we see that FT C_ T . Also lemrna 1 .2(ii) implies T C FT . Hence FT =T . (b) Let T' denote the class of torsion free abelian groups . Then T' is S-closed . It is trivial to see that FT' = T' . Suppose AftFT' . Since the only torsionfree factor group of a torsion abelian group is 0 we see that any N < t(AJ) is a direct summand of 1Vl (here t(M) denotes tire torsion subgroup of M) . It follows that any N <_ t(M) is a direct summand of t(M) and that t(M) itself is a direct summand of M . Thus t(M)cC and AJ = t(M) ® L with LcT' . This yields FT' C_ C® T' . Also AcC <~--> A = t(A) and t p (A) is a vector space over Z p for every prime p . Let M = A® B with AcC and BET' . Let N <AL Then t(N) < t(1V1) = A . Since AEC we get A = t(N) ® L and 101 8  K . VARADARAJAN both t(N) and L will be in C . From M = A® B= t(N) ® L ®B and N/t(N)cT_' we see that McI'T' . Hence C ®T' C_ I'T' . Using the reverse inclusion already proved we get FT' = CT T' . From lemma 1 .1(iv) we have FT' C OT' . We will actually give a complete characterization of the class áT' frorn which it will follow irnrnediately that the inclusion FT' C_ áT' is a strict inclusion . Let M6AT ' . Suppose for some prime p, the p-primary torsion t p (M) of M is non-zero . Then there exists a copy of Z p in t p (M) . Suppose N <_ M satisfies Z p + N = M . Either N f1 Z p = Z p or N n Z p = 0, in the former case N = M and in the latter case M = N ® Z p . If for all N <_ M satisfying Z p + N = M we have N = M, then Z p « M and this contradices the assumption that McAT' . Hence M = Z p ® N for some N <_ M . Thuswe have shown that if t p (M) :7É 0, any copy of Z p in t p (M) is a direct summand of M . In particular this implies that there are no elements of order p 2 in t p (M), hence t p (M) is a vector space over Z p . Hence t(M) =® p t p (M) is in C . We claim that (4)  n T' = {MeZ-mod / any Z p < M for any prime p is a direct summand of M} . Because of the observations in the earlier paragraph, to prove (4) we have only to show that if MeZ-mod has the property mentioned in the right hand side of (4) and if N « M then NcT' . If on the contrary there is an N « M with N 0 T', then t p (N) ,-É 0 for some prime p . Then there is a copy of Z p in t p (N) . Since N « M it will follow that this copy of Z p is small in M . However, any Z p < M being a direct summand of M cannot be small in M . From (4) we see that (direct product over all primos) is in AT' . However, t(M) =® p Z p and it is well-known that t(M) does not split off frorn M . Hence 111 « FT' . This proves that the inclusion FT' C áT' is strict . 2 . Study of AX when X = A n H, For results on dual Goldie dimension or corank the reader may refer to [7], [111 . As already remarked in [11], if the dual Goldie dimension ON CER .TAIN CLASSES OF MODULES  101 9 of M is infinito we cannot assert that there exists a surjéctive map cp M -> 11' 1 Ni with each Ni :,A 0 . (See Proposition 1.6 in [11]) . All we can assert in this case is that, given any integer d >_ 1 we can find a certain surjection 9 : M --> Il,q1 Lj with each Lj 7~ 0 (the modules L .- in general will depend on d ) . This different behaviour of dual Goldie dimension as compared to Goldie dimension necessitates many changos in the formulation and in the proofs of r esults dual to those obtained in Section 2 of [4] where the theory of Goldie dimension plays a crucial role . We first observe that the class H kis Q-closed . Lemma 2 .1 . Let X be Q-closed with X C_ H k . Let AllcOX and N _< M satisfy N + J(A11) = AJ . Assione that Al has property (P1) . Then M/NcH k . Proof ., Let rl : Al -> M/N denote the quotient map . From N + J(M) = M we get rl(J(M)) = M/N . Hence J(M/N)= M/N . Suppose if possible that M/N has dual Goldie dimension > k . Then there exists a surjection cp : M/N ---> A1 x . . . x Ae with 2 > k and each A j :y~ 0 . From J(M/N) = M/N we get J(A ;) = Aj for 1<_ j <_ Q . Since J(Aj) = Aj ,-á 0 and J(A ;) is the sum of all small submodules of A j we see that there exists a Bj « Aj with B . :~ 0 . Then B 1 x . . . x Bg « A 1 x . . . x Ae . From lemma 1 .6 we get A1 x . . . x A e cOX . This implies B 1 x . . . x BQcX . This contradicts the assumption that X C_ H k , since corank B 1 x . . . x BQ > 2 > k . Corollary 2 .2 . Suppose X is Q-closed and _X C_ H k . Let McáX . Suppose M Izas property (P1) and, sati .sfies,I(M) = M . Then McH k . Proof :: Choose N = 0 in lemma 2 .1 . Proposition 2 .3 . Let X be Q-closed with X C_ H k . Let Mci1X and assume that M has property (P1) . Then there exists an N6X such that 114'/N = B e H with B¿C and HEH k n OX . Proof .: Let L be a supplement of . .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 get LcOX . From LnJ(M) « L we get Ln J(M)cX . Since M/J(M) has property (P1) (Proposition 6 .1 in [13]) and J(M/J(M)) = 0 from proposition 3 .3 in [11] we see that M/J(M)cC . Hence L/(LnJ(M))EC . From lemma 2 .1 we get MILc_H k . lf we set N = LnJ(M) we get NeX and 0 -> L/N -> M/N -> MIL -> 0 exact . 102 6  K . VARADARAJAN We will now characterize the class V(T') . We will show that V(! :') = {McTI J(M) = 0} . Let McV(T') . We will show that 0 is the only small submodule of M . Then it follows that J(M) = 0 . Suppose on the contrary 0 :,A N « M . Since V(T') C_ T' we have MeT' . Hence NcT' . This means there is a copy of Z in N . Consider the subgroup 2Z of Z . From2Z <_ Z <_ N « 111 we get 2Z « M . Now, M/2Z has non-zero 2 torsion, contradicting the fact that McV(T') . Conversely any MeT' with J(M) = 0 is clearly in V(T') because then 0 is the only small submodule of M and M/0 - MeT' . This proves (5) . From (5) we see that the inclusion V(! :') C T' is a strict inclusion ; because Q6T' but Q « V(T') since J(Q) = Q . We included information en the classes LT, VT, LT' and VT' to complete the examples discussed in 1 .7 . References 1 .  A . W . CHATTERS ; A characterization of right Noetherian rings, Quarterly J . Math . Oxford 33 (1982), 65-69 . 2 .  DINH VAN HUYNH AND PHAN DAN, On rings with restricted minimum condition, Arch . Math . 51 (1988), 313-326 . 3 .  DINH VAN HUYNH ; NGUYEN V . DUNG AND PATRICK F . SMITH, Rings characterized by therr right ideals or cyclic modules, Proceedings of the Edinburgh Mathematical Society 32 (1989), 355-362 . 4 .  PATRICK F . SMITH, DIN VAN HUYNH AND NGUYEN V . DUNG, A characterization of Noetherian modules, Quart . J . Math . Oxford 41 (1990),225-235 . 5 .  DIN VAN HUYNH, NGUYEN V . DUNG, AND PATRICK F . SMITH, A characterization of rings with Krull dimension, J . Alg . 132 (1990), 104-112 . 6 .  DINH VAN HUYNH AND NGUYEN V . DUNG, A characterization of artinian rings, Clasgow Math . J . 30 (1988), 67-73 . 7 . B . SARATH AND K . VARADARAJAN, Dual Goldie dimension II, Communications in Alg . 7 (1979), 1885-1899 . 8 .  P . F . SMITH, Some rings which are characterized by finitely generated modules ; Quart . J . Math . Oxford 29 (1978), 101-109 . 9 .  P . F . SMITH, Rings characterized by therr cyclic modules, Canadian J . Math . 30 (1978), 98-111 . 10 .  P . VAMOS, The dual of the notion of finitely generated, J . London Math . Soc . 43 (1969) . ON CERTAIN CLASSES OF MODULES  1027 11 . K . VARADARAJAN, Dual Goldie dilnension, Communications in Alg . 7 (1979), 565-610 . 12 . K . VARADARAJAN, Modules with supplements, Pac . J . of Math . 82 (1979),559-564 . 13 . K . VARADARAJAN AND P . R . WANI, Modules over endomorphism rings 11, Acta lllath . HungaHca 53 (1989), 309-337 . 14 . K . VARADARAJAN, Hopfian and co-Hopfian objects (to appear) . Department of Mathematics and Statistics The University of Calgary 2500 University Drive N .W . Calgary, Alberta LANADA T2N 1N4 Primera versió rebuda el 15 de Novembre de 1991, darrera versió rebuda el 2de Mar4 de 1992