关于相对平坦性与相对凝聚性的注记

- 1 -A Note on Relative Flatness and Coherence 1Xiaoxiang Zhang Jianlong ChenDepartment of Mathematics Southeast University ，Nanjing 210096E-mail: z990303@AbstractLet R be a ring and M a fixed right R -module. A new characterization of M -flatness is given by certain linear equations. For a left R -module F such that the canonical map M ⊗R F →Hom R (M *, F ) is injective, where M * = Hom R (M , R ), the M -flatness of F is characterized via certain matrix subgroups. An example is given to show that R need not be M -coherent even if every left R -module is M -flat. Moreover, some properties of M -coherent rings are discussed. Keywords: M -flat module, matrix subgroup ，M -coherent ring1 IntroductionsRecently, Dauns [2] introduced the notion of coherence of a ring R relative to a right R -module M .Let R be a ring, M a fixed right R -module and σ[M ] the full subcategory of the category of right R -modules subgenerated by M (see [6, p. 118]). Recall from [2] that a left R -module F is σ[M ]-flat if for any exact sequence 0 → X → Y in σ[M ], the sequence 0 → X ⊗R F → Y ⊗R F is exact. Following Wisbauer [6], F is called M -flat if the sequence 0 → K ⊗R F → M ⊗R F is exact for every submodule 0 ≤ M . Dauns [2, Proposition 1.6] proved that R F is M -flat if and only if it is σ[M ]-flat.Following Dauns [2], a right R -module N is M -coherent if for any 0 ≤ A < B ≤ N such that B /A ⊆ mR for some m ∈ M , if B /A is finitely generated, then B /A is finitely presented. R is defined to be M -coherent if the right R -module R R is M -coherent.The purpose of this note is to investigate M -flat modules and M -coherent rings from some new aspects.Throughout R is an associative ring with identity and all modules are unitary. For a positive integer n , R n (resp. R n ) denotes the direct sum of n copies of R R (resp. R R ) whose elements are written as “row (resp. column) vectors”. Similarly, M n stands for the direct sum of n copies of M R . For each m = (m 1, m 2, …, m n ) ∈ M n , the right annihilator of m in R n is symbolized by)(m n R r , that is,)(m n R r = {(r 1, r 2, …, r n )T∈ R n | (m 1, m 2, …, m n ) (r 1, r 2, …, r n )T=∑=ni i i r m 1= 0}.Note that )(m n R r is nothing more than m ⊥ in [2] for all m ∈ M . Given an R -module P , the dual module and the character module are denoted by P * and P + respectively, i.e., P * = Hom R (P , R ) and P += Hom Z (P , Q /Z ), where Q (resp. Z ) is the ring of rational numbers (resp. integers).1This work is supported in part by the NSFC of China (No. 10571026) and the Foundation of Graduate Creative Program of JiangSu (xm04-10) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P.R.China.- 2 -2 Main resultsLet us start with the following result.Theorem 1. Let M be a fixed right R -module . The following are equivalent for a left R -module F .(1) F is M -flat .(2) For any a 1, …, a n ∈ R , and x 1, …, x n ∈ F such thatF m x a R ni ii )(1r ∈∑=for some m ∈ M ,there exist y 1, …, y k ∈ F and n ×k matrix B = (b ij )n ×k over R such that x i = ∑=kj j ijy b 1 for each 1 ≤ i ≤n and m∑=ni iji b a 1= 0 for each 1 ≤ j ≤ k .Proof . (1) ⇒ (2). For any a 1, …, a n ∈ R , and x 1, …, x n ∈ F , ifF m x a R ni ii )(1r ∈∑=, we haver R (m ) ⊆ L = a 1R + … + a n R + r R (m ) ≤ R Rwith L/r R (m ) finitely generated and.)(/)(0)(1F m LF F m F m x a R R R ni ii r r r ∈+=+∑=It follows by [2, Theorem 1.8 (d)] that.))(/())((1F m L x m a R i R n i i ⊗∈⊗+∑=r rLet a = (a 1 + r R (m ), …, a n + r R (m )) and let {e 1, …, e n } be the canonical basis of the free right R -module R n . There is an exact sequence0)(/(0→⎯→⎯⎯→⎯→m L R a R n R n r r ηαwhere α is the inclusion map and η is given by η (e i ) = a i + r R (m ) (i = 1, …, n ). Tensoring by F yieldsthe following exact sequence0)](/[(id id →⊗⎯⎯→⎯⊗⎯⎯→⎯⊗⊗⊗F m L F R F a R n R FF n r r ηαwhere id F : F → F is the identity map. Since,)](/[0))(())(id (11F m L x m a x e R ni i R i n i i i F ⊗∈=⊗+=⊗⊗∑∑==r r η∑=⊗ni i ix e 1∈ Ker(η⊗id F ) = Im(α⊗id F ). So we haveb 1 = ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡111n b b M , …, b k = ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡nk k b b M 1∈(a n R r and y 1, …, y k ∈ F such that∑∑==⊗=⊗ni i i kj j j x e y b 11∈ R n ⊗F . But∑∑∑∑∑=====⊗=⊗=⊗kj ni j ij i kj ni j ij i kj j j y b e y b e y b 11111)(])[(.)]([)()(111111∑∑∑∑∑∑======⊗=⊗=⊗=n i kj j ij i n i kj j ij i k j n i j ij i y b e y b e y b eHence x i =∑=⊗kj j j y b 1 for each i . Finally, it is easy to see that m ∑=ni ij i b a 1= 0 for each 1 ≤ j ≤ k .- 3 -(2) ⇒ (1). Suppose that L /r R (m ) is finitely generated with m ∈ M andr R (m ) ⊆ L = a 1R + … + a n R + r R (m ) ≤ R R .Then each element in (L /r R (m ))⊗F is of the form∑=⊗+n i i R i x m a 1))((rwith x i ∈ F (i = 1, …, n ). If∑=⊗+n i i R i x m a 1))((r is contained in the kernel of the natural mapµ: (L /r R (m ))⊗F → LF /r R (m )Fi.e.,F m x a R ni ii )(1r ∈∑=, we have y 1, …, y k ∈ F and n ×k matrix B = (b ij )n ×k over R such that x i =∑=kj j ij y b 1 for each 1 ≤ i ≤ n and m ∑=ni ij i b a 1= 0 for each 1 ≤ j ≤ k by (2). Consequently,∑=⊗+ni i R i x m a 1))((r =)())((11∑∑==⊗+ni kj j ij R i y b m a r=)))(((11∑∑==⊗+n i k j j R ij i y m b a r =]))([(11∑∑==⊗+k j ni j R ij i y m b a r =∑=⊗+k j j R y m 1))(0(r = 0 ∈ (L /r R (m ))⊗F .This shows µ is monic and hence F is M -flat by [2, Theorem 1.8 (d)].Note that a left R -module F is M -flat if and only if 0 → K ⊗R F → M ⊗R F is exact for every finitely generated submodule K of M (see [6, 12.15(1)]. Let K =∑=n i i R m 1and f : R n → M be the compositionof the canonical epimorphism π : R n → K and the inclusion map α: K → M . Applying Hom R (−, R ) to f ,we obtain f *: M * → R n with imageIm f * = {(g(m 1), g (m 2), …, g (m n )) | g ∈ M * = Hom R (M , R ) },which is a matrix subgroup of R n in sense of Zimmermann [7]. We refer the reader to [7] for the general case. Following Zimmermann [7], we write m = (m 1, …, m n ) ∈ M n and H M ,m (R ) = {(g(m 1), g (m 2), …, g (m n )) | g ∈ M *}.Then we have the following commutative diagram with exact bottom row)*,(Hom )),((Hom 0)(,id 1F M F R H FM F R m R m M R R FR ni i →→↓↓⊗⎯⎯→⎯⊗⊗=∑ψϕαwhere ψ : M ⊗R F → Hom R (M *, F ) is given byψ (m ⊗y ): g a g (m )yfor all m ∈ M , y ∈ F and g ∈ M *, and ϕ: (∑=n i i R m 1) ⊗R F → Hom R (H M ,m (R ), F ) is defined such that,)())(,),(),((:)(1211∑∑==⊗ni i i n ni i i y m g m g m g m g y m a L ϕfor all y i ∈ F (i = 1, …, n ) and g ∈ M *. If ϕ is injective then so is α⊗id F and the converse holds in caseψ is injective. So we have the followingProposition 2. Let M be a right R -module and F a left R -module such that the canonical map ψ : M ⊗R F- 4 -→ Hom R (M *, F ) is injective , then the following are equivalent for a left R -module F .(1) F is M -flat .(2) The canonical map ϕ : (∑=n i i R m 1) ⊗R F → Hom R (H M ,m (R ), F ) is injective for all positiveintegers n and all m = (m 1, …, m n ) ∈ M n .Remark 3. It should be pointed out that the hypothesis “M ⊗R F → Hom R (M *, F ) is injective” is essential for (1) ⇒ (2) in Proposition 2. To see this we need only take a finitely generated right R -module M =∑=n i i R m 1which is not torsionless. Then F = R R is M -flat but it does not satisfy (2) incase m = (m 1, …, m n ) ∈ M n . In fact, there exist 0 ≠ m = ∑=n i i i r m 1∈ M and g ∈ M * such that g (m ) = 0since M is not torsionless. If follows that )(1∑=⊗ni i ir m ϕ= 0 but ∑=⊗ni i i r m 1 1)(1⊗∑=ni i i r m = m ⊗ 1 ≠ 0 in M ⊗R .Observing the following commutative diagram with exact rows)),((Hom ),(Hom )),(/(Hom 00)()(,,1F R H F R F R H R F R m F R F m m M R n R m M n R R ni i R n R R n →→→↓≅↓↓→⊗⎯→⎯⊗⎯→⎯⊗∑=ϕσ rwhere σ is the canonical map such that σ(a ⊗y ): b + H M ,m (R )a bay , for all a ∈)(m n R r , y ∈ F and b∈ R n , we have the following proposition.Proposition 4. Let M be a right R -module and F a left R -module . Then the following are equivalent for every m = (m 1, …, m n ) ∈ M n .(1) The canonical map ϕ : (∑=n i i R m 1) ⊗R F → Hom R (H M ,m (R ), F ) is injective .(2) The canonical map σ :(m n R r ⊗R F → Hom R (R n /H M ,m (R ), F ) is surjective .(3) Every f ∈ Hom R (R n /H M ,m (R ), F ) factors through a finitely generated free R -module , i.e., there exists R R k (1 ≤ k ∈ N ), g ∈ Hom R (R n /H M ,m (R ), R k ) and h ∈ Hom R (R k , F ) such that f = hg .(4) R n /H M ,m (R ) is projective with respect to every short exact sequence of the form0 → R L → R P → R F → 0 (∗)i.e., (∗) is R n /H M ,m (R )-pure in sense of Rothmaler [5].Proof . (1) ⇔ (2) follows from the commutative diagram mentioned above.(2) ⇒ (3). For each f ∈ Hom R (R n /H M ,m (R ), F ), by (2), there exist b j = [b 1j , …, b nj ]T ∈)(m n R r (1 ≤ j ≤ k ) and y 1, …, y k ∈ F such that σ (kj 1=Σb j ⊗y j ) = f . Now, define g ∈ Hom R (R n /H M ,m (R ), R k ) via g (a + H M ,m (R )) = aB for all a ∈ R n , where B = (b 1, …, b k ), and define h ∈ Hom R (R k , F ) via h (e j ) = y j for all 1 ≤ j ≤ k , where {e 1, …, e k } is the canonical basis of R k . It is easy to verify that h and g are as desired.- 5 -(3) ⇒ (2). Suppose (3), then for each f ∈ Hom R (R n /H M ,m (R ), F ), we have f = hg for some h ∈Hom R (R k , F ) and g ∈ Hom R (R n /H M ,m (R ), R k ). Let B =⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡++))(())((,,1R H g R H g m M n m M εεM , where {ε1, …, εn } is thecanonical basis of R n , and let y j = h (e j ), where {e 1, …, e k } is the canonical basis of R k . It follows that σ (kj 1=Σb j ⊗y j ) = f , where b j is the j -th column of B (1 ≤ j ≤ k ). Therefore σ is surjective.(3) ⇒(4) follows by the following commutative diagram(4) ⇒ (3). Applying (4) to the exact sequence 0 → R L → R R (I )→ R F → 0 where I is a suitableindex set, we have the following commutative diagramSince R n /H M ,m (R ) is finitely generated, the image of f is contained in a finitely generated free submodule R k of R R (I ). Then (3) follows easily.Corollary 5. Let M be a right R -module and F a left R module such that the canonical map ψ : M ⊗R F → Hom R (M *, F ) is injective , then the following are equivalent .(1) F is M -flat .(2) The canonical map σ : )(m n R r ⊗R F → Hom R (R n /H M ,m (R ), F ) is surjective for all positive integers n and all m = (m 1, …, m n ) ∈ M n .(3) For all positive integers n and all m = (m 1, …, m n ) ∈ M n , every R -homomorphism from R n /H M ,m (R ) to F factors through a finitely generated free R -module .(4) Every exact sequence of the form 0 → R L → R P → R F → 0 is R n /H M ,m (R )-pure for all positive integers n and all m = (m 1, …, m n ) ∈ M n .Next, we consider M -flatness for factor modules of M -flat modules.Proposition 6. Suppose that M is a fixed right R -module and L is a submodule of an M -flat left R -module F . If the canonical map (M /K )⊗L →(M /K )⊗F is injective for each (finitely generated ) K R < M R then F /L is M -flat . The converse holds if the canonical map M ⊗L → M ⊗F is injective .Proof . Let us denote M /K and F /L by M and F respectively. Consider the following exact commutative diagram for each (finitely generated) K R < M R- 6 -FM F M d d F K F K F M L M F M F M d d F M L M F M F M d R d R d R d R ⊗⎯→⎯⊗↓↓⊗⎯→⎯⊗↓↓⊗⎯→⎯⊗→⎯→⎯↓↓↓↓⊗⎯→⎯⊗→⎯→⎯ 621151114387),(Tor ),(Tor ),(Tor ),(Torwhere d 2 is injective since F is M -flat by hypotheses and hence d 1 is surjective. If d 4 is injective then d 3 is surjective and so is d 5. It follows that d 6 is injective. This shows that F is M -flat.Conversely, if d 6 and d 8 are injective then d 5 and d 7 are surjective. Consequently, d 3 is surjective and hence d 4 is injective.Remark 8. Note that the canonical map M ⊗L → M ⊗F in the above proposition need not be injective in general even if every left R -module is M -flat. Recall that a ring R is right FS (see [4]) if the socle of R R is flat as a right R -module. Now, let us take a ring R which is not right FS (e.g., the ring R in Example 11 below) and M = Soc(R R ), the socle of R R . Obviously, every left R -module is M -flat since M is semisimple. But M is not flat hence M ⊗L → M ⊗F need not be injective in general.Corollary 9. Let L be a pure submodule of an M -flat left R -module F then both L and F /L are M -flat .Proof . Let L be a pure submodule of an M -flat left R -module F . Then for every submodule K of M , we have the following commutative diagramFM L M g h FK L K f⊗⎯→⎯⊗↓↓⊗⎯→⎯⊗ where f and g are monic and hence so is h . Therefore L is M -flat. Finally, F /L is M -flat by Proposition 6.Given a ring R and a fixed right R -module M . Let M -F (respectively F ) be the class of all M -flat (respectively flat) left R -modules. Then we haveProposition 10. (1) R is von Neumann regular if and only if M -F = F for every right R -module M .(2) R is right noetherian if and only if R is M -coherent for every (cyclic ) right R -module M .Proof . (1) is obvious.(2). If R is right noetherian then R is right coherent and r R (m ) is finitely generated for all m ∈ M . Hence R is M -coherent by [2, Observations 2.5 (i)]. Conversely, for each right ideal I of R , R is R /I -coherent and hence I = r R (1+I ) is finitely generated. Therefore R is right noetherian.It is well known that R is right coherent if and only if every direct product of flat left R -modules is- 7 -flat. The following example shows that the analogous statement for M -flatness and M-coherence fails. It also shows that the hypothesis “r R (m ) is finitely generated for all m ∈ M ” is essential for [2, Theorem 2.6] and answers the question in [2, p. 308].Example 11. There is a ring R with a module M R such that every left R -module is M -flat but R is not M -coherent .Proof . Note that every left R -module is M -flat if and only if every submodule M is pure in M , i.e., M is regular in Mod-R in sense of Wisbauer [6]. Let Z 2 = Z /2Z and A =∞=⊕1i Z 2 be the direct sum of countably infinite copies of Z 2. Then,R = Z 2 ∝ A = {⎥⎦⎤⎢⎣⎡a a 0α| a ∈ Z 2, α ∈ A },the trivial extension of Z 2 by A is a commutative ring with Soc(R ) = 0 ∝ A . Now, let M = Soc(R ). It follows that every left R -module is M -flat since M is semisimple and hence regular in Mod-R . But theright annihilator ofm = ⎥⎦⎤⎢⎣⎡00),0,0,1(0Lis not finitely generated. Therefore R is not M-coherent by [2, Consequences 2.3 (1)].Note that M = Soc(R ) is the unique maximal (right) ideal of the ring R in Example 11. Moreover, the left annihilator of M in R is M itself. By [4, Theorem 2.4], R is not (right) FS as we claimed in Remark 8.Let L be a left R -module and C a class of left R -modules. Recall from [3] that an R -homomorphism ϕ : L → C with C ∈ C is called a C –preenvelope of L if every ψ ∈ Hom R (L , C') with C' ∈ C factors through ϕ. The class C is said to be preenveloping if every left R -module has a C -preenvelope.Theorem 12. Let M be a right R -module . Then(1) M -F is closed under direct products if and only if M -F is preenveloping .(2) R Γ is M -flat for any set Γ if and only if every projective right R -module P has an M -flat dual module P * if and only if every direct product of flat left R -modules is M -flat .Proof . (1). By Corollary 9, M -F is closed under pure submodules. Thus (1) follows by a slight modification of the proof of [3, Proposition 6.5.1].(2). For every projective right R -module P , we have P ⊕ Q = R (I ) for some Q R and indexed set I . It follows that P * ⊕ Q * = (R (I ))* = R I . If R I is M -flat then so is P *. Conversely, suppose every projective right R -module P has an M -flat dual module P *. Then for every index set I , the free right R -module R (I ) has an M -flat dual (R (I ))* = R I .To complete the proof, it remains only to show that every direct product of flat left R -modules is M -flat provided R Γ is M -flat for any set Γ.For any set {F γ | γ ∈ Γ} of flat left R -modules, we have pure exact sequences- 8 -)(00)(Γ∈→→→→γγγγ F RK Iand00)(→Π→Π→Π→Γ∈Γ∈Γ∈γγγγγγF RK Iwhere)(γγI RΓ∈Π is a pure submodule of γγI RΓ∈Π. But γγI RΓ∈Π is M -flat by hypothesis.Therefore)(γγI R Γ∈Π is M -flat by Corollary 9. Since )(0γγγγI RK Γ∈Γ∈Π→Π→is pure,γγF Γ∈Π is M -flat by Corollary 9 again.Remark 13. So far it is unknown to us whether M -F is closed under products whenever R Γ is M -flat for any set Γ. To find a ring R with a right R -module M such that R Γ is M -flat for any set Γ but M -F is not closed under products, we have to consider those rings which are neither right noetherian nor von Neumann regular. Moreover the module M should not be regular in Mod-R .But by [2, Theorem 2.6], we haveCorollary 14. The following are equivalent for a fixed right R -module M such that r R (m ) is finitely generated for all m ∈ M .(1) R is M -coherent .(2) M -F is preenveloping .(3) P * ∈ M -F for every projective right R -module P .Recall from [6] that a right R module U is M -injective if every diagram of right R -modules with exact rowUMK ↓→→0can be extended commutatively by a morphism M → U . It is proved by Dauns [2, Theorem 1.7] that a left R -module F is M -flat if and only if the character module F + = Hom Z (F , Q /Z ) is M -injective. The following result is motivated by [1, Theorem 1].Proposition 15. Consider the following conditions for a fixed right R -module M :(1) A right R -module N is M -injective if and only if N + is M -flat .(2) A right R -module N is M -injective if and only if N ++ is M -injective . (3) A left R -module F is M -flat if and only if F ++ is M -flat . (4) M -F is closed under direct products . We have (1) ⇔ (2) ⇒ (3) ⇒ (4).Proof . (1) ⇔ (2) ⇒ (3) follows from [2, Theorem 1.7].(3) ⇒ (4). We adopt the method in [1, Theorem 1]. For any index set I and F i ∈ M -F (i ∈ I ), we have- 9 -)()()(0→⊕⎯→⎯Π⎯→⎯⊕↓Π⎯→⎯Π⎯→⎯++∈++∈≅++∈≅++∈∈i I i g i I i i I i i I i fi I i F F F F Fwhere ⊕i ∈I F i is an M -flat left R -module. Then (⊕i ∈I F i )++ is M -flat by (3). Since ⊕i ∈I +i F is a pure submodule of ∏i ∈I +i F by [1, Lemma 1(1)], it follows that g is split epic. Consequently, (⊕i ∈I +i F )+ is M -flat and so is ∏i ∈I ++i F . By [1, Lemma 1(2)], f is pure monic. This guarantees that ∏i ∈I F i is an M -flat left R -module.Remark 16. Note that (3) of Proposition 15 does not implies (1) even if R is M -coherent. For instance, the endomorphism ring R of a countably infinite dimensional vector space V over a field is von Neumann regular. Let M = R R then (3) of Proposition 15 holds since every left R -module F is (M -)flat. By [1, Theorem 2], (1) of Proposition 15 does not hold since R is not right noetherian.References[1] T. J. Cheatham & D. R. Strone (1981) Flat and projective character modules. Proc . Amer . Math .Soc ., 81: 175-177[2] J. Dauns (2006) Relative flatness and coherence. Comm . Algebra , 34: 303-312[3] E. E. Enochs & O. M. G. Jenda (2000) Relative Homological Algebra . de Gruyter Expositions inMath. 30. de Gruyter, Berlin[4] Z. K. Liu (1995) Rings with flat left socle. Comm . Algebra , 23: 1645-1656[5] P. Rothmaler (1994) Mittag-Leffler modules and positive atomicity . Manuscript, Kiel[6] R. Wisbauer (1991) Foundations of Module and Ring Theory . Philadelphia: Gordon and BreachSience Publishers[7] W. Zimmermann (1997) Modules with chain conditions for finite matrix subgroups. J . Algebra ,190: 68-87关于相对平坦性与相对凝聚性的注记张小向 陈建龙东南大学数学系，（210096）摘 要：本文利用某些特定的方程组给出了M -平坦性的一个新的刻画，其中M 为环R 上的一个固定的模。