ON CERTAIN CLASSES OF MODULES

ON CERTAIN CLASSES OF MODULES


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 when we 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 .
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, Ka 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 .8respectively 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 module and 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. (i) Straight forward .
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] .As in [4] when the ring R is clear from the context, M, Z, P, I, C, G, N, A, U; Ktt 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 > Nl x . . .x N,., 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 Hk .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.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 n N) ; hence L n N -N%KeI ®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==> Lemma 1.2.Let X be a class of R-modules .Then (i) FX, ~X, FX are all S-closed.
(ii) If X is S-closed, then X C_ FX and XC C_ OX .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 Zp2 1 OZ for any prime p.This shows that áZ is not Q-closed .
Before proceeding further we need to recall some definitions and results from [7], [111,J12] ; [13] 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 (Pl ) 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 (Pi) for i = l, 2 (Proposition 6.20 in [13] and Proposition 2 .29 in [11]) .Clearly P2 => 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 FT.Combining this with the S-closed nature of FT 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 tp (A) is a vector space over Zp 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 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' = C T 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 tp (M) of M is non-zero .Then there exists a copy of Zp in tp (M) .Suppose N <_ M satisfies Zp + N = M. Either N f1 Zp = Zp or N n Zp = 0, in the former case N = M and in the latter case M = N ® Zp.If for all N <_ M satisfying Zp + N = M we have N = M, then Zp « M and this contradices the assumption that McAT' .Hence M = Zp ® N for some N <_ M. Thus we have shown that if tp(M) :7É 0, any copy of Zp in tp (M) is a direct summand of M. In particular this implies that there are no elements of order p2 in tp (M), hence tp (M) is a vector space over Z p .Hence t(M) = ®p tp(M) is in C.
We claim that (4) n T' = {MeZ-mod / any Zp < 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 tp (N) ,-É 0 for some prime p. Then there is a copy of Zp in tp (N) .Since N « M it will follow that this copy of Zp is small in M.However, any Zp < 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 Zp 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 .

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 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,q-1 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 Hk is Q-closed .Lemma 2.1 .Let X be Q-closed with X C_ Hk .Let AllcOX and N _< M satisfy N + J(A11) = AJ .Assione that Al has property (P1) .Then M/NcHk .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 Aj :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 Aj we see that there exists a Bj « Aj with B. :~0.Then B 1 x . . .x Bg « A1 x . . .x Ae .From lemma 1.6 we get A1 x . . .x AecOX .This implies B1 x . . .x BQcX.This contradicts the assumption that X C_ Hk , since corank B1 x . . .x BQ > 2 > k.Proof.: Let L be a supplement of ..I(A4) in AJ.Then L + J(M) = M and L n J(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 L n 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 .
Since L + J(M) = M any xEM can be written as ex + ux with e,,EL and ux EJ(M) .If x = ex + ux = e x + uy are two such expressions, we have e x -e~= ux -ux is in L n j(m) = N.Thus the element ex in L/N representad by ea; depends only on x.Moreover, if xcN, x = 0 + x is such an expression, hence e x = 0 .It follows that one gets a welldefined map n : M/N --> L/N given by a(x + N) = ex + N. It is easily checked that a yields a splitting of the inclusion L/N --> M/N .Hence M/N -L/N ® MIL .Moreover L/NeC and M/LcHk n AX.
That M/LEOX is a consequence of lemma 1.6.Set B = L/N and H=M/L .Proposition 2.4.Let X be a {P, Q, S}-closed class of R-modules.Let McR-mod and N <_ M satisfy NcX and M/N = B®(H1+---+Hk) with BEC and Hl + --+ Hk an irredundant sum of hollow modules (this sum need not be direct) .Suppose HicAX for 1 < i < k .Then McAX .
Proof.Let K « M. We have to show that Kcx.Let rl : M -> M/N denote the quotient map.Writing H for Hl + ---+ Hk we have M/N = B ® H. Since K « M we get K < J(M), hence r7(K) < r1(J(M)) < J(M/N) = J(H) since BcC .
(i) It is clear that any non-artinian module M will contain a proper non-artinian submodule .Hence if M is a module with the property that NEA for all N ~M then M itself is in A. In particular a hollow module H will satisfy HcOA if and only if HeA .(ii) The classes A and N are {P, Q, S}-closed .Hence proposition 2 .4 is valid when X = A or N.
(iii) Any AlleA or any MeC or any hollow module M has property (P2) .
Modules with finito spanning dimension in the sense of P. Fleury (Section 4 of [11]) have property (P2) .All artinian modules have finito spanning dimension and hence finite corank .
The following results proved in [7], [ll] will be needed later in our present paper (iv) If McH k has property (P2) then AJ can be written as an irredundant sum Hl + ---+ H,. of hollow modules with r <_ k .This is Theorem 2.39(1) in [11] .(v) If M = Hl + ---+ H,. with Hi hollow, then corank M < ,r .This is Proposition 1 .7 in [7] .For this part we need not assume that M has property (P2).
Let us denote the class of modules with property (P¡) by Mi (i = 1, 2) .We can state one of our main results as follows .(b) The class A n Hk is Q-closed .Let M,12 n 0(A n _Hk ) .From proposition 2 .3,there exists an N6A n H k such that M/N = B ® II with BEC and HcH k n 0(A n Hk) .Since All has (P2 ) it follows from proposition 2 .29 in [11] that H has (P2) .Hence H = Hi + ---+ H,. an irredundant sum of hollow modules with r _< k .Rom lemrna 1. .2(i) each IIj is in á(AnH k ) .In particular IhcA(_A) .From remark 2 .5(i)we see that Hj cA .Thus H6A n Hk by remark 2 .5) .This proves (b).Stated in words Theorem 2.6(b) takes the following forro.Theorem 2 .7.Let Al be a module with property (P2) .Suppose every small submodule of M is artinian and of dual Coldie dimension < k.Then there exists an artinian submodule N of M with corank N <_ k such that M/N = B ® L with B semi-simple and L artinian of corank < k .Corollary 2.8 .Suppose Al is a module with property (P2 ) and of finite corank .Suppose every small submodule of M is artinian and of dual Coldie dimension < k for some fixed integer k .Then M is artinian.Proo£ From the above theorerri, there exists an artinian submodule N of M such that M/N = B ® L with BEC and LcA.No-,v, corank B <_ corank M/N < corank M < oo .A semi-simple module has finite corank if and only if it is semi-simple artinian.It follows that BcA and hence MEA .
We have a variant of corollary 2.8 which is actually easier to prove.Proposition 2.9.Let M be a finitely generated module with property (P1) .Then Mc n A if and only if M is artinian.
Proof.: Since M/J(M) has property (P1) and J(M/J(M)) = 0 it follows that M/J(11)cC .Since M is finitely generated it follows that M/J(M) is semi-simple artinian.Since M is finitely generated we also have J(M) « M. Thus HEAA =* J(M)cA .From 117/J(M)EA we get MEA .Conversely, we have already observed that A C AA .Proof.We will abbreviate finitely generated as f.g and finitely embedded as f.e.We have M/J(M)6C because M/J(M) has (P1) and J(M/J(M)) = 0.If we show that M/J(M) is f.g it will follow from S C_ X and the P-closed nature of X that M/J(111)cX .Again J(M)eX and M/J(M)EX will yield MeX .
Suppose on the contrary M/J(M) is not f.g.Then M/J(M) = Vi ®V2 with Vi, V2 semi-simple and each not f.g.Since a non f.g semi-simple module does not have finito corank we see that Vi « X for i = 1, 2 .Let 91 : .M -> M/J(M) denote the quotient map and L1 = rl-1(V1) .Since V1 1 X and X is Q-closed it follows that L1 ~X .From 1V1cFX we get M = N1 ® Wi with Nr <_ L1 and L1 /NI eX.From L1 1 X we get Ni 0. From J(M) = J(N1 ) ® J(4V1 ) we get j(m) n N1 = J(NI ) and J(M) n GV1 = J(W1 ) .This yields MIJ(M) = (Ni /J(N,)) (Wi/J(W1)).Also N1 /J(N1 ) = Nl/J(M) nNi <_ L1 /J(M) = Vi .Since M/J(M) = Vi ®V2 and V2 is not f.g and Nr/J(N1) < V1 it follows that W1/J(W1) is not f.g.Since Wr is a direct surnmand of M we see that W1 is f.e.Since Wi is a quotient of M we see that 6V1 has property (P1) .From lemma 1.2(i), since VV1 <_ M we get WicFX .Since X is Q-closed, from J(M)eX we get J(W1 )cX .Thus W1 satisfies all the conditions imposed on M and further TVl / J(W1 ) is not f.g.Hence the same arguments as above will yield a decomposition Wl = N2 ®W2 with N2 0 0, W2 f.e with property (P1 ), W2 EFX, J(W2)EX and W2/J(W2) semi-simple but not f.g.Iteration of this argument yields for any integer k >_ 1 a direct sum decomposition M = Ni ® . . .® Nk ® Wk with each Nj 5 E 0. This means that the Goldie dimension of M >_ k for every integer k > l .However, any Le module trivially has finite Goldie dimension .This contradiction shown that M/J(M) has to be f.g thus completing the proof of proposition 2.10.

. Dual of Chatters' result
As stated in the introduction Chatters has proved that if every cyclic R-module is a direct sum of a projective module and a noetherian module, then R is noetherian .The module theoretic generalization obtained in [4] asserted that G n DN = N.In this section we will prove two forros of duals for the above mentioned result.Theorem 3.1 .Let McR-mod satisfy the condition that every submodule of M is the direct sum of an injective module and an artinian module.Suppose further that M satisfies one of the following conditions : (i) M has (Pl ) and is f..g or (ii) M has (P1 ), is f. e and J(117) « M .
Then MeA .
Proof.Part of our hypothesis could be rephrased as MEF(I®A) .Since A is {S, P}-closed, from lernma 1 .2(iv)we infer that Me(I ® A) n FA.
In case (i) is valid, proposition 2.9 immediately yields MeA.In case (ii) is valid, the assumption that J(M) « M implies that J(M)EA.Then proposition 2 .10yields MEA .
Conversely, if MEA every N < M satisfies NEA .Thus N = 0 ® N is an expression for N as the direct sum of an injective module and an artinian module .Corollary 3 .2.Let R be a semi-perfect ring.Then every left ideal of R is a direct sum of an injective left ideal and an artinian left ideal if and only if R is left artinian.
Proof: This is an immediate consequence of theorem 3 .1(i) .

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) .

(
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).
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 -L n NEX.Also MeI ==> KeI; hence NEI ® X.This means M6F(I ® X) yielding I n FX C 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 < Nl < < 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) .

Theorem 2 . 6 .
We have following inclusions.(a) (C ® A)A C DA.(b) M2 n ~(A n Hk) C (CEDA n Hk)A n Hk .Proof. .(a) Let Me(C ® A)A.Then there exists an N _< M with NCA and M/N = B ® L with BcC and L6A .Since L has (P2) and of finito dual Coldie dimension we can write L = II, + ---+ H,. an irredundant sum of hollow modules (see iv) in remark 2 .5) .From LcA we see that Hi,EA.Since A is S-closed, we have A C_ AA.From proposition 2 .4we see that McAA .

Proposition 2 .
10 .Suppose X is a {P, Q}-closed class satisfyinq S C X C H. Suppose M is a finitely embedded module with property (P1) satisfying McFX and J(M)eX .Then McX .