summary:Recently Rim and Teply [11] found a necessary condition for the existence of $\sigma$-torsionfree covers with respect to a given hereditary torsion theory for the category $R$-mod. This condition uses the class of $\sigma$-exact modules; i.e. the $\sigma$-torsionfree modules for which every its $\sigma$-torsionfree homomorphic image is $\sigma$-injective. In this note we shall show that the existence of $\sigma$-torsionfree covers implies the existence of $\sigma$-exact covers, and we shall investigate some sufficient conditions for the converse

Bican, Ladislav

Torrecillas, Blas

English

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeLadislav Bican; Blas TorrecillasRelative exact coversCommentationes Mathematicae Universitatis Carolinae, Vol. 42 (2001), No. 4, 601--607Persistent URL: http://dml.cz/dmlcz/119276Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 2001Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 42,4 (2001)601–607 601Relative exact coversLadislav Bican, Blas TorrecillasAbstract. Recently Rim and Teply [11] found a necessary condition for the existence ofσ-torsionfree covers with respect to a given hereditary torsion theory for the categoryR-mod. This condition uses the class of σ-exact modules; i.e. the σ-torsionfree modulesfor which every its σ-torsionfree homomorphic image is σ-injective. In this note we shallshow that the existence of σ-torsionfree covers implies the existence of σ-exact covers,and we shall investigate some sufficient conditions for the converse.Keywords: precover, cover, hereditary torsion theory σ, σ-injective module, σ-exactmodule, σ-pure submoduleClassification: 16D90, 16S90, 18E40In what follows, R stands for an associative ring with identity and R-moddenotes the category of all left unital R-modules. A class G of modules is calledabstract if it is closed under isomorphic copies. Recall that a homomorphism ϕ :G→M is called a G-precover of the moduleM if G ∈ G and every homomorphismf : F → M , F ∈ G, factors through ϕ, i.e. there exists g : F → G suchthat ϕg = f . Moreover, a G-precover ϕ of M is said to be a G-cover , if eachendomorphism f of G such that ϕf = ϕ is the automorphism of the module G.As usual, a hereditary torsion theory σ = (T ,F) for the category R-mod con-sists of two abstract classes T and F , the σ-torsion class and the σ-torsionfreeclass , respectively, such that Hom(T, F ) = 0 whenever T ∈ T and F ∈ F . Theclass T is closed under submodules, factor modules, extensions and direct sums.The class F is closed under submodules, extensions and direct products. For eachmoduleM there exists an exact sequence 0→ T →M → F → 0 such that T ∈ Tand F ∈ F . To each hereditary torsion theory σ it is associated the Gabriel filterL of left ideals consisting of all left ideals I ≤ R with R/I ∈ T . Recall that σ issaid to be of finite type if L contains a cofinal subset L′ of finitely generated leftideals. A module F is called σ-injective if it is injective with respect to all exactsequences E : 0→ A→ B → C → 0, where C ∈ T . Baer Test Lemma states thatit suffices to consider the sequence E where B = R or, equivalently, the sequencesThis work has been initiated while the first author was visiting the University of Almeŕıa.The first author has been partially supported by the Grant Agency of the Czech Republic,grant #GAČR 201/98/0527 and also by the institutional grant MSM 113 200 007.The second author has been partially supported by PB98-1005 from DGES.602 L. Bican, B.Torrecillaswhere B = R and A ∈ L. The class of all σ-torsionfree σ-injective modules willbe denoted by J . Following [11], we say that a σ-torsionfree module is σ-exactif any its σ-torsionfree homomorphic image is σ-injective. The class of all suchmodules will be denoted by E .The investigations of this paper are motivated by several sources. First, if σis an exact torsion theory for R-mod, i.e. a hereditary torsion theory such thatE(M)/M ∈ J whenever M ∈ F and M ∈ J , then J = E and it is well-known(see e.g. [12] or [6]) that every module has a J -cover whenever σ is of finitetype, i.e. if σ is perfect . Further, it is known that a sufficient condition for theexistence of F -covers (σ-torsionfree covers) is equivalent to the condition that thedirected unions of σ-torsionfree σ-injective modules are σ-injective (see [12] and[9; Proposition 43.9]). On the other hand, recently [11] presented a necessarycondition saying that the directed union of σ-exact modules is σ-injective. Someother examples having similar motivating character are mentioned at the end ofthe paper.So, the purpose of this note is to investigate and unify the methods and relationsbetween σ-torsionfree and σ-exact covers under several hypotheses on the torsiontheories or classes of modules, respectively.Recall from [4] that a submodule N of a moduleM is said to be G-pure, G beingan abstract class of modules, if the factor module M/N belongs to the class G.Further, it is well-known (and not too hard to verify) that a σ-torsionfree precoverϕ : G → M of a module M is the σ-torsionfree cover of M if and only if Ker ϕcontains no non-zero submodule σ-pure (i.e. F -pure) in G. The following resultsare well-known, we include these assertions here for the sake of completeness.1. Lemma. Let E : 0→ Ai−→ Bπ−→ C → 0 be an exact sequence. Then(i) if B ∈ E and C ∈ F then A,C ∈ E ;(ii) if A,C ∈ E then B ∈ E .2. Remark. If F ∈ E , then a submodule K of F is E-pure in F if and onlyif it is σ-pure (i.e. σ-closed) in F . This simple fact follows immediately fromthe definitions but, since it will be frequently used in the sequel, it seems to beconvenient to formulate it here explicitly. A similar situation occurs with the factthat if an abstract class G is closed under finite direct sums (or, more generally,under extensions) and under directed unions, then it is closed under arbitrarydirect sums. Clearly, let F =⊕λ∈Λ Fλ, Fλ ∈ G, and let {Kω|ω ∈ Ω} be thecollection of all finite subset of the set Λ. Setting Fω =⊕λ∈KωFλ, we see thatFω ∈ E and the union F =⋃ω∈Ω Fω is directed in the natural way.Now we are going to show that the existence of σ-torsionfree covers impliesthat the class E is closed under directed unions and that for each F ∈ E the setof all E-pure submodules is closed under directed unions, too. The first assertionslightly generalizes [11; Theorem 1].Relative exact covers 6033. Proposition. Let σ = (T ,F) be a hereditary torsion theory for the categoryR-mod such that every module has a σ-torsionfree cover. Then(i) the class E is closed under directed unions;(ii) for an arbitrary module F ∈ E the set of all E-pure submodules of F isclosed under directed unions.Proof: (i) Let F =⋃λ∈Λ Fλ be a directed union of σ-exact modules. We firstshow that F ∈ J . If ϕ : G→ Eσ(F )/F is an F -cover of Eσ(F )/F = σ(E(F )/F ),then for each λ ∈ Λ we have the commutative diagramEσ(F )/Fλ Eσ(F )/FλfλyyπλGϕ−−−−→ Eσ(F )/Fwhere πλ is the canonical projection and fλ exists by the definition of a precoverin view of the fact that Eσ(F )/Fλ is σ-torsionfree, Fλ being σ-injective. Clearly,Ker fλ ⊆ F/Fλ and we show that the equality Ker fλ = F/Fλ holds for eachλ ∈ Λ. Assuming that Ker fλ 6= F/Fλ for some λ ∈ Λ, we can find an indexµ ∈ Λ such that fλ(Fµ+FλFλ) 6= 0. Now fλ(Fµ+FλFλ) is σ-exact as the σ-torsionfreehomomorphic image of FµFλ∩Fµand so it is σ-pure in G. Obviously, fλ(Fµ+FλFλ) ⊆Ker ϕ, which contradicts by Remark 2 the fact that ϕ is an F -cover of Eσ(F )/F .Thus Ker fλ = F/Fλ; hence Im fλ ∼= Eσ(F )/F , Eσ(F )/F is σ-torsionfree andF = Eσ(F ) is σ-torsionfree σ-injective.Now let K ≤ F be any σ-closed submodule. Then F/K =⋃λ∈ΛFλ+KK is adirected union of σ-exact submodules Fλ+KK∼= FλFλ∩Kand so it is σ-injective bythe first part of the proof.(ii) Let K =⋃λ∈ΛKλ be a directed union of E-pure submodules of the moduleF ∈ E . The only property we need to show is that K is σ-closed in F . However,Kλ is, as a σ-closed submodule of F , σ-exact for each λ ∈ Λ by Lemma 1 and, con-sequently, K is σ-exact by (i). Hence K is σ-closed in F by [9; Proposition 10.1].Similarly to the case of σ-torsionfree covers (Remark 2), we have the followingresult for E-covers.4. Proposition. Let σ = (T ,F) be a hereditary torsion theory for the categoryR-mod. An E-precover ϕ : G→M of a moduleM is an E-cover of M if and onlyif Ker ϕ contains no non-zero submodule E-pure in G.Proof: Assume first that ϕ is an E-cover of M , and let K ⊆ Ker ϕ be an E-pure604 L. Bican, B.Torrecillassubmodule of G. Consider the following diagramGπ−−−−→ G/K̺−−−−→ Gϕyyϕ̄yϕM M Mwhere π is the canonical projection and ϕ̄ is the corresponding natural map. SinceG/K ∈ E , the definition of a precover yields the existence of a homomorphism̺ : G/K → G such that ϕ̺ = ϕ̄. So, ϕ(̺π) = ϕ̄π = ϕ gives that ̺π is anautomorphism of G; hence π is injective and K = 0.Conversely, let the condition be satisfied and let f be an endomorphism ofthe module G such that ϕf = ϕ. Obviously, Ker f ⊆ Ker ϕ and so G/Ker f ∼=Im f ∈ F yields by Remark 2 that f is injective. Further, the submodule H ={u− f(u) |u ∈ G} of G is clearly contained in Ker ϕ and as an epimorphic imageof G it is σ-pure in G by [9; Proposition 10.1]. Thus H = 0 and f is surjective.Following [4], we say that an abstract class G of modules satisfies condition(P ), if to each infinite cardinal λ there exists a cardinal κ > λ such that for everyF ∈ G with |F | ≥ κ and everyK ≤ F with |F/K| ≤ λ, the submodule K containsa non-zero submodule L such that F/L ∈ G. In [4; Theorem 2 and Corollary 3] ithas been proved that if an abstract class G of modules is closed under direct sums,satisfies condition (P ) and, for each F ∈ G, the set of all G-pure submodules ofF is inductive, then every module has a G-precover. If, in addition, the class G isclosed under direct limits, then every module has a G-cover.5. Theorem. Let σ = (T ,F) be a hereditary torsion theory for the categoryR-mod. If every module has a σ-torsionfree cover, then every module has anE-cover.Proof: The class E is closed under arbitrary direct sums by Proposition 3(i)and Remark 2. The set of all E-pure submodules of each member of E is induc-tive by Proposition 3(ii) and E satisfies condition (P ) by [6; Theorem 11]. Thusevery module has an E-precover by [4; Theorem 2]. Moreover, in view of Propo-sition 3(ii) we can find an E-precover ϕ : G → M of the module M such thatKer ϕ contains no non-zero E-pure submodule of G. Thus ϕ is an E-cover of Mby Proposition 4. 6. Corollary. Let σ = (T ,F) be a hereditary torsion theory of finite type forthe category R-mod. Then(i) the class E is closed under directed unions;(ii) the set of all E-pure submodules of any member of E is closed underdirected unions;(iii) every module has an E-cover.Relative exact covers 605Proof: By [12; Theorem] and [5; Corollary 4.1], every module has a σ-torsionfreecover and it suffices to apply Proposition 3 and Theorem 5. Now we proceed to formulate some conditions that are sufficient to prove theconverse statement to Theorem 5. Note that the hypotheses of Proposition 8 aremotivated by the ordinary torsion theory on the category Ab of all abelian groups.7. Proposition. Let σ = (T ,F) be a hereditary torsion theory for the categoryR-mod.(i) If 0 6= F =⋃λ∈Λ Fλ is a directed union of members of the class E and ifF has an E-cover, then F belongs to the class E .(ii) If K =⋃λ∈ΛKλ is a directed union of E-pure submodules of a moduleF ∈ E such that K has an E-cover, then K is E-pure in F .Proof: (i) For an E-cover ϕ : G→ F of the module F , consider the diagramFλιµλ−−−−→ Fµ̺µ−−−−→ G∥∥∥∥∥∥yϕFλιµλ−−−−→ Fµιµ−−−−→ Fwhere 0 6= Fλ ⊆ Fµ and ιµλ, ιµ are the inclusion maps. By hypothesis, for eachλ ∈ Λ there exists ̺λ : Fλ → G such that ϕ̺λ = ιλ. From this we infer that ϕ 6= 0and since F is obviously σ-torsionfree, Ker ϕ is σ-pure in G. So, by Remark 2,it is E-pure in G. Thus ϕ is injective by Proposition 4 and so ϕ(̺λ − ̺µιµλ) =ιλ − ιµιµλ = 0 yields ̺λ = ̺µιµλ. This means that ̺µ extends ̺λ wheneverFλ ⊆ Fµ and, consequently, there naturally exists a homomorphism ψ : F → Gsuch that ψιλ = ̺λ for each λ ∈ Λ. By the same argument ϕψιλ = ϕ̺λ = ιλgives ϕψ = 1. Thus ϕ is an isomorphism and hence F ∈ E .(ii) Each Kλ, λ ∈ Λ, is σ-pure in F by [9; Proposition 10.1] and so it lies inthe class E by Lemma 1. By (i) we have that K ∈ E ; hence K is σ-pure in F by[9; Proposition 10.1] again. Remark 2 finishes the proof. 8. Proposition. Let σ = (T ,F) be a hereditary torsion theory for the categoryR-mod. If F is a σ-torsionfree σ-injective module such that every non-zero σ-torsionfree factor module and every submodule of F that is expressible as adirected union of members of E has a non-zero E-cover, then F is σ-exact.Proof: Let ϕ0 : G0 → F be an E-cover of the module F . We may assumeF 6= 0, the case F = 0 being trivial. By the hypothesis and the unicity of coverswe have ϕ0 6= 0. Since G0/Ker ϕ0 ∼= Im ϕ0 ∈ F , ϕ0 is a monomorphism byProposition 4 and Remark 2. Hence K0 = ϕ0(G0) is a non-zero submodule of Flying in the class E . For K0 6= F assume that for some ordinal α the submodulesKβ , β < α, of F have already been constructed in such a way that Kγ ⊂ Kγ+1606 L. Bican, B.Torrecillasfor each γ + 1 < α and Kβ ∈ E for each β < α. For a limit ordinal α we simplyset Kα =⋃β<αKβ, and we have Kα ∈ E by the hypothesis and Proposition 7(i).If α = β + 1 is a successor ordinal then, similarly to the beginning, we take anE-cover ϕβ : Gβ → F/Kβ and we denote ϕβ(Gβ) =Kβ+1Kβ= KαKβ . Then Kβ ⊂ Kαand Kα/Kβ ∈ E , which together with Lemma 1 yields that Kα ∈ E . ThusF =⋃α<λKα for some ordinal λ and F ∈ E by Proposition 7(i), again. 9. Theorem. Let σ = (T ,F) be a hereditary torsion theory for the categoryR-mod such that Qσ(R) ∈ E . The following conditions are equivalent:(i) σ is of finite type;(ii) every module has a σ-torsionfree cover;(iii) every module has an E-cover.Proof: (ii) follows from (i) by [12; Theorem] in the faithful case and by [5;Corollary 4.1] in the general case. Further, (iii) follows from (ii) by Theorem 5and we are going to show that (i) follows from (iii). So, let J = σ(R), I ∈ L bearbitrary and I =⋃λ∈Λ Iλ be a directed union of a finitely generated left idealof R. For each λ ∈ Λ we take the σ-closure Kλ ofIλ+JJ in Eσ(R/J) = Qσ(R).It is easy to see that for Iλ ⊆ Iµ we have Kλ ⊆ Kµ and so, taking the directedunion K =⋃λ∈ΛKλ we get thatI+JJ ⊆ K. However, from the isomorphism(R/J)/((I+J)/J) ∼= R/(I+J) we infer that (I+J)/J is σ-dense in K, R/(I+J)being σ-torsion as the homomorphic image of R/I. Consequently, (I + J)/J isσ-dense in Eσ(R/J). On the other hand, Kλ is E-pure in Eσ(R/J) for each λ ∈ Λby the hypothesis and so Proposition 7(ii) yields that K is E-pure in Eσ(R/J).Thus K = Eσ(R/J) and hence there is an index λ ∈ Λ such that 1 + J ∈ Kλ,which means that R/J ⊆ Kλ and soIλ+JJ is σ-dense in R/J . Finally, consideringthe exact sequence 0→ Iλ+JIλ→ RIλ→ RIλ+J→ 0, we see that R/(Iλ+ J) ∈ T bythe above part. Since Iλ+JIλ∼= JJ∩Iλ ∈ T obviously, we have Iλ ∈ L, as we had toshow. 10. Remarks. We conclude by some simple applications of the above theoryand we also present some examples motivating this theory, as promised at thebeginning.It follows easily from Corollary 6 that if σ = (T ,F) is a hereditary torsiontheory of finite type for the category R-mod over a left semihereditary ring R,then the classes J and E coincide and, consequently, every module has a σ-torsionfree σ-injective cover. Further, as a consequence of Theorem 9 one canobtain that an exact torsion theory σ = (T ,F) is perfect if and only if everymodule has an J -cover. Since J = E in this case, σ is perfect if and only if everymodule has an E-cover.If σ is the ordinary torsion theory on the category Ab of all abelian groups,then Eσ(Z) = Qσ(Z) = Q is σ-cocritical and, consequently, it belongs to the classRelative exact covers 607E trivially. In this case J = E is the class of all torsionfree divisible groups andso every module has an E-cover by Theorem 9, σ being obviously of finite type.Further, if σ is any hereditary torsion theory for the category Ab of all abeliangroups, then it is well-known that σ is the P -torsion for a suitable subset P ofthe set of all primes Π. Then Eσ(Z) consist of all rationals with denominatorsdivisible by the primes from P only, and it is an easy exercise to verify thatany σ-torsionfree homomorphic image of Eσ(Z) is either Eσ(Z) itself, or it is afinite groups of order relatively prime to each p ∈ P . Such groups are obviouslyσ-injective and so Eσ(Z) belongs to the class E .Finally, let R be an arbitrary commutative domain and σ be the ordinarytorsion theory on the category R-mod. By [9; Corollary 44.3] σ is exact as theGoldie’s torsion theory. Since σ is obviously of finite type, it is perfect and,consequently, every module has a σ-torsionfree injective cover in view of the simplefact that for each F ∈ F the factor module E(F )/F is σ-torsion and, consequently,the class J consists just of σ-torsionfree injective modules.References[1] Anderson F.W., Fuller K.R., Rings and Categories of Modules, Graduate Texts in Mathe-matics, vol.13, Springer-Verlag, 1974.[2] Bican L., El Bashir R., Enochs E., All modules have flat covers, Bull. London Math. Soc.33 (2001), 385–390.[3] Bican L., Kepka T., Němec P., Rings, Modules, and Preradicals, Marcel Dekker, New York,1982.[4] Bican L., Torrecillas B., On covers, J. Algebra 236 (2001), 645–650.[5] Bican L., Torrecillas B., Precovers, to appear.[6] Bican L., Torrecillas B., On the existence of relative injective covers, to appear.[7] Enochs E., Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981),189–209.[8] Garćıa Rozas J.R., Torrecillas B., On the existence of covers by injective modules relativeto a torsion theory, Comm. Algebra 24 (1996), 1737–1748.[9] Golan J., Torsion Theories, Pitman Monographs and Surveys in Pure an Applied Mathe-matics, 29, Longman Scientific and Technical, 1986.[10] Rada J., Saoŕın M., Rings characterized by (pre)envelopes and (pre)covers of their modules,Comm. Algebra 26 (1998), 899–912.[11] Rim S.H., Teply M.L., On coverings of modules, to appear.[12] Teply M., Torsion-free covers II, Israel J. Math. 23 (1976), 132–136.[13] Torrecillas B., T-torsionfree T-injective covers, Comm. Algebra 12 (1984), 2707–2726.[14] Xu J., Flat covers of modules, Lecture Notes in Mathematics, 1634, Springer Verlag, Berlin-Heidelberg-New York, 1996.Department of Algebra, Charles University, Faculty of Mathematics and Physics,Sokolovská 83, 186 75 Prague 8, Czech RepublicE-mail : bican@karlin.mfff.cuni.czDepartment of Algebra and Analysis, Universidad de Almeŕıa, 04071 Almeŕıa, SpainE-mail : btorreci@ual.es(Received November 13, 2000, revised September 3, 2001)

Relative exact covers

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Relative exact covers

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Institute of Mathematics AS CR, v. v. i.