τ-complemented and τ-supplemented modules

Journal Algebra Discrete Math. Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2006). pp. 1 – 15 c Journal “Algebra and Discrete Mathematics” τ-complemented and τ-supplemented modules Khaled Al-Takhman, Christian Lomp, Robert Wisbauer Abstract. Proper classes of monomorphisms and short exact sequences were introduced by Buchsbaum to study relative homological algebra. It was observed in abelian group theory that complement submodules induce a proper class of monomorphisms and this observations were extended to modules by Stenstr¨om, Generalov, and others. In this note we consider complements and supplements with respect to (idempotent) radicals and study the related proper classes of short exact sequences. 1. Proper classes and τ-supplements With the intention of formalising the theory of Ext functors depending on a specific choice of monomorphisms, Buchsbaum introduced in [4] certain conditions on a class of monomorphisms which are needed to study relative homological algebra. This lead to he notion of proper classes of monomorphisms and short exact sequences. Well-known examples of such classes are the pure exact sequences which can be defined by choosing a class Pof (finitley presented) modules and considering those sequences on which Hom(P, −)is exact for each P∈ P. These techniques are outlined, for example, in Mishina and Skornjakov [10], Sklyarenko [13] and [16]. It was observed in abelian group theory that complement submodules induce a proper class of short exact sequences and this motivated the investigation of such questions for modules over arbitrary rings. First results in this direction were obtained, for example, by Stenstr¨om [14] and Generalov [5, 7]. For a more comprehensive presentation of the results and the sources we refer to E. Mermut’s PhD thesis [9]. 2000 Mathematics Subject Classification: 16D90, 16E50, 16.E99. Key words and phrases: proper classes; neat and coneat submodules; τsupplemented, τ-complemented and τ-semiperfect modules. Journal Algebra Discrete Math. 2Complemented and supplemented Our interest in this approach to the structure theory of modules is based on an observation mentioned in Stenstr¨om [14], namely that over any ring, supplement submodules induce a proper class of short exact sequences. This may help to bring some order in the variations and generalisations of supplemented and lifting modules coming up recently by involving properties from torsion theory. 1.1. Proper classes. Let Ebe a class of short exact sequences in σ[M]. If 0//Kf//Lg//N//0 belongs to E, then fis called an E-mono and gis said to be an E-epi. The class Eis called proper if it satisfies the conditions P.1 Eis closed under isomorphisms; P.2 Econtains all splitting short exact sequences in σ[M]; P.3 the class of E-monos is closed under composition; if f′, f are monos and f′◦fis an E-mono, then fis an E-mono; P.4 the class of E-epis is closed under composition; if g, g′are epis and g◦g′is an E-epi, then gis an E-epi. The class of all splitting short exact sequences in σ[M]is an example of a proper class. 1.2. Purities. Let Pbe a class of modules in σ[M]. Denote by EPthe class of all short exact sequences in σ[M]on which Hom(P, −)is exact for each P∈ P. It is straightforward to prove that EPis a proper class. This type of class is called projectively generated and its elements are called P-pure sequences. The "classical" purity is obtained by taking for Pall finitely presented modules in σ[M]; for M=Rthis is the Cohn purity (e.g., [16, § 33]). 1.3. Copurities. Let Qbe a class of modules in σ[M]. Denote by EQ the class of all short exact sequences in σ[M]on which Hom(−, Q)is exact for each Q∈ Q. Then EQis a proper class. This type of class is called injectively generated and its elements are called Q-copure sequences (see [16, § 38]). 1.4. Relative injectivity and projectivity. Let Ebe a proper class of short exact sequences in σ[M]. A module P∈σ[M]is called E-projective if Hom(P, −)is exact on all short exact sequences in E. Dually, a module Q∈σ[M]is called E-injective if Hom(−, Q)is exact on all short exact sequences in E. Journal Algebra Discrete Math. K. Al-Takhman, Ch. Lomp, R. Wisbauer 3 It follows from standard arguments that the class of all E-projective modules is closed under direct sums and direct summands, and the class of all E-injective modules is closed under direct products and direct summands. 1.5. Class of complement submodules. Let Ecbe the class of short exact sequences 0→K→L→N→0in σ[M]such that Kis a complement (closed) submodule of L. Then: (1) Ecis a proper class in σ[M]. (2) Every (semi-) simple module (in σ[M]) is Ec-projective. Proof. (1) The conditions P.1 and P.2 are easily verified. P.3. Let K⊆Land L⊆Nbe closed submodules. Choosing b K⊆ b L⊆b N, we have K=b K∩L=b K∩b L∩N=b K∩N, proving that Kis closed in N. Consider K⊆L⊆N. If Kis closed in N, then K=b K∩N=b K∩L, that is, Kis closed in L. P.4. The composition of two epimorphisms gand hcan be presented by the commutative diagram with exact rows and columns, 0//K =  //U i  //U/K //  0 0//K//L gh  g//L/K // h  0 L/U =//L/U, where U/K = Ke h, for some K⊆U⊆L. Assume gand hto have closed kernels. Suppose that U= Ke gh is not closed in Land denote by Uthe essential closure of Uin L. Then Kis closed in Uand Ke h=U/K EU/K, contradicting the assumption that hhas a closed kernel. On the other hand, assume U= Ke gh to be closed. Suppose that Ke hhas a proper essential extension Vin L/K. Then U= (Ke h)g−1E (V)g−1, contradiction our condition on Ke h. This shows that Ke his a closed submodule of L/K. (2) Let K⊆Lbe a closed submodule and S⊆L/K any simple submodule. Then S≃N/K for some submodule K⊆N⊆L. By Journal Algebra Discrete Math. 4Complemented and supplemented assumption, Kis a maximal submodule and is not essential in N. Hence the map N→N/K ≃Ssplits showing that Hom(S, L)→Hom(S, L/K) is surjective and that Sis Ec-projective. Since direct sums of Ec-projectives are again Ec-projective, every semisimple module is Ec-projective. 1.6. τ-complement submodules. Let τbe an idempotent preradical for σ[M]with associated classes Tτand Fτ. Then for a submodule K⊆L where L∈σ[M], the following are equivalent: (a) every N∈Tτis projective with respect to the projection L→L/K; (b) there exists a submodule U⊆Lsuch that K∩U= 0 and τ(L/K) = (U+K)/K ≃U; (c) there exists a submodule U⊆Lsuch that K∩U= 0 and τ(L/K)⊆(U+K)/K ≃U. If this conditions are satified, then Kis called a τ-complement in L. Proof. (a)⇒(b) A pullback construction yields the commutative diagram with exact rows 0//KiK// =  e K//  τ(L/K)// i  0 0//K//Lg//L/K //0. Since τ(L/K)∈Tτ, there exists a morphism h:τ(L/K)→Lwith i=hg. By the Homotopy Lemma (e.g., [16, 7.16]), this implies that the top row splits, that is, e K=K⊕U, for some U⊆e Kand (K+U)/K =τ(L/K). (b)⇒(a) Let N∈Tτand f∈Hom(N, L/K). Then Im f⊆τ(L/K) and it can be seen from the diagram in the proof of (a)⇒(b) that there is a morphism h:N→Lwith f=hg. (b)⇔(c) This is easy to verify. 1.7. Corollary. Let τbe a preradical for σ[M]and K⊆Lwhere L∈ σ[M]. (1) If Kis a τ-complement in Land L/K ∈Tτ, then Kis a direct summand. Journal Algebra Discrete Math. K. Al-Takhman, Ch. Lomp, R. Wisbauer 5 (2) If L∈Tτ, then every τ-complement submodule of Lis a direct summand. For the preradical induced by the class of all (semi-)simple modules we find an interesting relationship with the complement submodules. 1.8. Neat submodules. A monomorphism f:K→Lis called neat if any simple module Sis projective relative to L→L/Im f, that is, the Hom sequence Hom(S, L)→Hom(S, L/K)→0is exact. The class of short exact sequences with neat monomorphisms is a projectively generated class in the sense of 1.2. As shown in 1.5, all sequences in Ecare neat. 1.9. When neat submodules are closed in σ[M].For a module M the following are equivalent: (a) every neat submodule of Mis closed; (b) a submodule of Mis closed if and only if it is neat; (c) for every L∈σ[M], closed submodules of Lare neat; (d) for every essential submodule U⊆M,Soc M/U 6= 0; (e) every M-singular module is semi-artinian. Proof. (a)⇔(b) is clear since closed submodules are neat. (c)⇒(a) is obvious. (a)⇒(d) Let UEMbe a proper submodule. Then Uis not closed and hence not neat in M. Thus there exists a morphism g:S→M/U where Sis simple, that can not be extended to a morphism S→M. In particular, this implies that Im g6= 0, that is, Soc M/U 6= 0. (d)⇒(e) Let U⊆V⊆M. If UEMthen VEMand hence (d) implies that every factor module of M/U has nonzero socle, that is, M/U is semiartinian (see [6, 3.12]). By [6, Proposition 4.3], the set {M/U |UEM}is a generating set of all M-singular M-generated modules and every M-singular module is a submodule of M-generated M-singular modules. Thus if all the M/U have nonzero socles then this is also true for all M-singular modules. (e)⇒(c) Let K⊆Lbe a neat submodule and assume that it has a proper essential extension K⊆L. Then K/K is an M-singular module and hence, by assumption, contains a simple submodule S=N/K where KEN⊆K. Now neatness of K⊆Limplies that the map N→N/K = Ssplits. This contradicts KENproving that Kis closed in L. For M=Rwe obtain the following characterisation which was (partly) proved in [5, Theorem 5]: Journal Algebra Discrete Math. 6Complemented and supplemented 1.10. When neat submodules are closed in R-Mod.For a ring R the following are equivalent: (a) every neat left ideal of Ris closed; (b) a left ideal of Ris closed if and only if it is neat; (c) for every left R-module, closed submodules are neat; (d) for every essential left ideal I⊆R,Soc R/I 6= 0; (e) every singular module is semi-artinian. Rings with these properties are called C-rings (in [12]). Dualising the notions considered above yields the following. 1.11. τ-supplement submodules. Let τbe a radical for σ[M]with associated classes Tτand Fτ. Then for a submodule K⊆Lwhere L∈ σ[M], the following are equivalent: (a) every N∈Fτis injective with respect to the inclusion K→L; (b) there exists a submodule U⊆Lsuch that K+U=Land U∩K=τ(K); (c) there exists a submodule U⊆Lsuch that K+U=Land U∩K⊆τ(K). If this conditions are satisfied, then Kis called a τ-supplement in L. Proof. (a)⇒(b) Consider the commutative diagram with exact rows 0//Ki// p  L//  L/K // =  0 0//K/τ(K)//L/τ(K)//L/K //0. Since K/τ(K)∈Fτ, there exists h:L→K/τ(K)with p=ih. By the Homotopy Lemma (e.g., [16, 7.16]), this implies that the bottom row splits, that is, L/τ(K) = K/τ(K)⊕U/τ(K),for some τ(K)⊆U⊆L. This means L=K+Uand U∩K=τ(K). (b)⇒(a) By the given data, L/τ(K) = K/τ(K)⊕U/τ(K). Let N∈Fτand f∈Hom(K, N). Then τ(K)⊆Ke fand we have the Journal Algebra Discrete Math. K. Al-Takhman, Ch. Lomp, R. Wisbauer 7 commutative diagram 0//Ki// p  f                  L//  L/K // =  0 K/τ(K)// ¯ f zzv v v v v v v v v L/τ(K)//L/K //0. N Since the middle row splits we obtain a morphism h:L→Nwith f=ih. (b)⇔(c) One direction is trivial. Assume K+U=Land K∩U⊆τ(K). Putting U′=U+τ(K)we have K+U′=Land K∩U′=K∩U+τ(K) = τ(K). 1.12. Corollary. Let τbe a radical for σ[M]and K⊆Lwhere L∈ σ[M]. (1) If Kis a τ-supplement in Land K∈Fτ, then Kis a direct summand. (2) If L∈Fτ, then every τ-supplement submodule of Lis a direct summand. (3) If Kis a τ-supplement in Land X⊆K, then K/X is a τsupplement in L/X. Proof. (1) and (2) follow from the preceding observations. (3) Let U⊆L be such that K+U=Land K∩U⊆τ(K). Then K/X +(U+X)/X = L/X and K∩(U+X)/X = (K∩U+X)/X ⊆(τ(K) + X)/X ⊆τ(K/X). As a special case we consider the radical (for σ[M]) cogenerated by the simple modules. 1.13. Co-neat submodules. A monomorphism f:K→Lis called co-neat if any module Qwith Rad Q= 0 is injective relative to it, that is, the Hom sequence Hom(L, Q)→Hom(K, Q)→0is exact. The class of short exact sequences with co-neat monomorphisms is an injectively generated class in the sense of 1.3. Journal Algebra Discrete Math. 8Complemented and supplemented 1.14. Characterisation of co-neat submodules. For a submodule K⊆L, the following are equivalent: (a) K→Lis a co-neat submodule; (b) there exists a submodule U⊆Lsuch that K+U=Land U∩K= Rad K; (c) there exists a submodule U⊆Lsuch that K+U=Land U∩K⊆Rad K. If these conditions are satisfied, then Kis a Rad-supplement in L. If RadK≪K, then Kis co-neat (Rad-supplement) in Lif and only if it is a supplement in L(see [9]). 1.15 Lemma. A small submodule Nof a module Lis co-neat in Lif and only if Rad N=N. Proof. Let N≪L. If Nis co-neat in L, then there exists K⊆Lsuch that N+K=Land N∩K= Rad N. Since N≪L,K=Land hence N=N∩L= Rad N. On the other hand assume N≪Land Rad N=N. Then N+L=Land N∩L=N= Rad N, thus Nis a co-neat submodule of L. 1.16. When are co-neat submodules coclosed. Let Mbe a module. Then the following conditions are equivalent: (a) every non-zero co-neat submodule of a module in σ[M]is a coclosed submodule; (b) every non-zero M-small module in σ[M]is a Max module (resp. has a maximal submodule). Proof. (a)⇒(b) If N≪Land Rad N=N, then, by the Lemma 1.15, N is a co-neat submodule of L. But by hypothesis co-neat submodules are coclosed submodules and hence not small - a contradiction. (b)⇒(a) Let Nbe a co-neat submodule of a module L∈σ[M]. Then for any submodule U⊆N,N/U is co-neat in M/U. To see this let K be a submodule of Lsuch that N+K=Land N∩K=Rad(N). Then N/U + (K+U)/U =L/U and N/U (K+U)/U = ((N K) + U)/U = (Rad N+U)/U ⊆Rad N/U. Hence N/U is co-neat in L/U. Suppose N/U ≪L/U, then Rad N/U = N/U by the Lemma 1.15. But by hypothesis N/U is a Max module, and hence has a proper maximal submodule. Hence N/U 6≪ L/U for all U⊂Nimplies that Nis coclosed in L. Journal Algebra Discrete Math. K. Al-Takhman, Ch. Lomp, R. Wisbauer 9 Let Mbe a cosemisimple module. Then Rad N= 0 for any N∈ σ[M]. Hence any non-zero submodule of a module in σ[M]is coclosed. Moreover any non-zero module is a Max module. However a submodule Nof a module Lis co-neat if and only if it is a direct summand. Thus if Mis not semisimple, there are coclosed submodules which are not coneat. This shows that in general the dual statement of the statement for neat submodules (1.9 (b)⇔(e)) does not hold. 2. τ-supplemented modules Throughout this section τwill denote a radical for σ[M]. Recall that a submodule K⊆Lis called a τ-supplement provided there exists some U⊆Lsuch that U+K=Land U∩K⊆τ(K)(1.11). To some extent the theory of supplemented, lifting and semiperfect modules can be transferred to the corresponding notions based on τ-supplements. This will be sketched in this section. 2.1. Definition. A module Lis said to be τ-supplemented if every submodule K⊆Lhas a τ-supplement in L, and it is called amply τsupplemented if for any submodules K, V ⊆Lsuch that K+V=L, there is a τ-supplement Ufor Kwith U⊆V. 2.2. τ-supplemented modules. Let Lbe a τ-supplemented module in σ[M]. (1) Every submodule K⊆Lwith K∩τ(L) = 0 is a direct summand. In particular, if Lis τ-torsion-free, then Lis semisimple. (2) Every factor module and every direct summand of Lis τ-supplemented. (3) L/τ(L)is a semisimple module. (4) L=U⊕Nwhere Nis semisimple and τ(U)EU. Proof. (1) Recall that τ(K)⊆K∩τ(L)and then refer to 1.12. (2) and (3) are also obvious consequences of 1.12. (4) Let N⊆Lbe a complement for τ(L), i.e. N∩τ(L) = 0 and N⊕τ(L)EL. This implies τ(N) = 0. By assumption, there exists U⊆Lsuch that N+U=Land N∩U⊆τ(U). By construction, N∩U=N∩(N∩U)⊆N∩τ(U)⊆N∩τ(L) = 0, hence L=N⊕Uand τ(L) = τ(N)⊕τ(U) = τ(U). Thus N⊕τ(U)EN⊕U and this implies τ(U)EU. By (1), Nis semisimple. 2.3. Sums of τ-supplemented modules. Let L∈σ[M]. Journal Algebra Discrete Math. 16 Complemented and supplemented Robert Wisbauer Mathematisches Institut der, HeinrichHeine Universit¨at, 0225 D¨usseldorf, Germany E-Mail: [email protected] Received by the editors: 16.01.2006 and in final form 21.11.2006.