Rings which have (m,n)-flat injective modules

A ring R is said to be a left IF − (m, n) ring if every injective left R-module is (m, n)-flat. In this paper, several characterizations of left IF − (m, n) rings are investigated, some conditions under which R is left IF−(m, n) are given. Furthermore, conditions under which a left IF −1 ring (i...

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Datum:	2005
Hauptverfasser:	Zhanmin, Z., Zhangsheng, X.
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Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2005
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spelling	irk-123456789-1573332019-06-21T01:26:15Z Rings which have (m,n)-flat injective modules Zhanmin, Z. Zhangsheng, X. A ring R is said to be a left IF − (m, n) ring if every injective left R-module is (m, n)-flat. In this paper, several characterizations of left IF − (m, n) rings are investigated, some conditions under which R is left IF−(m, n) are given. Furthermore, conditions under which a left IF −1 ring (i.e., IF −(1, 1) ring) is a field, a regular ring and a semisimple ring are studied respectively. 2005 Article Rings which have (m,n)-flat injective modules / Z. Zhanmin, X. Zhangsheng // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 4. — С. 93–100. — Бібліогр.: 12 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16D50, 16E65. http://dspace.nbuv.gov.ua/handle/123456789/157333 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	A ring R is said to be a left IF − (m, n) ring if every injective left R-module is (m, n)-flat. In this paper, several characterizations of left IF − (m, n) rings are investigated, some conditions under which R is left IF−(m, n) are given. Furthermore, conditions under which a left IF −1 ring (i.e., IF −(1, 1) ring) is a field, a regular ring and a semisimple ring are studied respectively.
format	Article
author	Zhanmin, Z. Zhangsheng, X.
spellingShingle	Zhanmin, Z. Zhangsheng, X. Rings which have (m,n)-flat injective modules Algebra and Discrete Mathematics
author_facet	Zhanmin, Z. Zhangsheng, X.
author_sort	Zhanmin, Z.
title	Rings which have (m,n)-flat injective modules
title_short	Rings which have (m,n)-flat injective modules
title_full	Rings which have (m,n)-flat injective modules
title_fullStr	Rings which have (m,n)-flat injective modules
title_full_unstemmed	Rings which have (m,n)-flat injective modules
title_sort	rings which have (m,n)-flat injective modules
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2005
url	http://dspace.nbuv.gov.ua/handle/123456789/157333
citation_txt	Rings which have (m,n)-flat injective modules / Z. Zhanmin, X. Zhangsheng // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 4. — С. 93–100. — Бібліогр.: 12 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT zhanminz ringswhichhavemnflatinjectivemodules AT zhangshengx ringswhichhavemnflatinjectivemodules
first_indexed	2025-07-14T09:46:39Z
last_indexed	2025-07-14T09:46:39Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 4. (2005). pp. 93 – 100 c© Journal “Algebra and Discrete Mathematics” Rings which have (m, n)-flat injective modules Zhu Zhanmin, Xia Zhangsheng Communicated by V. M. Futorny Abstract. A ring R is said to be a left IF − (m,n) ring if every injective left R-module is (m,n)-flat. In this paper, several characterizations of left IF − (m,n) rings are investigated, some conditions under which R is left IF−(m,n) are given. Furthermore, conditions under which a left IF −1 ring (i.e., IF − (1, 1) ring) is a field, a regular ring and a semisimple ring are studied respectively. 1. Introduction Throughout R is an associative ring with identity and all modules are unitary. The character module HomZ(M, Q/Z) of a module M will be denoted by M∗ and the injective hull of M by E(M). Let m, n be two fixed positive integers. Let G an abelian group. We write Gn(Gn) for the set of all formal n-dimensional row(column) vectors over G. An R-module M is said to be n-generated if it has a generating set of cardinality at most n[6]. A left R-module M is called (m, n)- injective if for every n-generated submodule K of the left R-module Rm, each R-homomorphism from K to M can be extended to Rm[3]. Note that (1, n)-injective modules are also called n-injective in [10],[12]. The ring R is left (m, n)-injective if RR is (m, n)-injective. It is obvious that RM is FP -injective if and only if RM is (m, n)-injective for all positive integers m and n, and RM is P-injective if and only if RM is (1, 1)- injective. We also recall that a module M is FP -injective if and only if M is absolutely pure, M is flat if and only if M∗ is FP -injective. A submodule U ′ of a module RU is pure in U if and only if the canonical 2000 Mathematics Subject Classification: 16D50, 16E65. Key words and phrases: injective modules; (m, n)-flat modules; left IF−(m, n) rings; left IF − 1 rings. Jo u rn al A lg eb ra D is cr et e M at h .94 Rings which have (m, n)-flat injective modules map V ⊗R V → V ⊗R U is monic for every finitely presented module VR. Indicated by these results we have following definition. Definition 1.1. Let m, n be two fixed positive integers. (1) A right R-module V is said to be (m, n)-presented if there exists an exact sequence of right R-modules 0 → K → Rm → V → 0 with n-generated K. (2) Given a left R-module U , a submodule U ′ of U is called (m, n)- pure in U if for every (m, n)-presented right R-module V , the canonical map V ⊗R U ′ → V ⊗R U is monic. (3) A left R-module M is (m, n)-flat if for every n-generated submod- ule K of the right R-module Rm, the canonical map K⊗RM → Rm⊗RM is monic. (4) A module M is called (ℵ0, n)-injective ((ℵ0, n)-flat) if M is (k, n)- injective ((k, n)-flat) for all positive integers k. Module M is called (m,ℵ0)-injective if M is (m, k)-injective for all positive integers k. A submodule U ′ of a module U is said to be (ℵ0, n)-pure ((m,ℵ0)-pure) in U if U ′ is (k, n)-pure ((m, k)-pure) in U for all positive integers k. An exact sequence 0 → U ′ → U → M → 0 is called (m, n)-pure if U ′ is (m, n)-pure in U , and a module L is called (m, n)-pure projective if L is projective with respect to every (m, n)-pure exact sequence. Using a standard technique we can prove the following theorem. Theorem 1.2. (1) An exact sequence of left R-modules 0 → U ′ → U → M → 0 is (m, n)-pure if and only if every (n, m)-presented module is projective with respect to this exact sequence if and only if (U ′)m ⋂ KU = KU ′ for every n-generated submodule K of the right R-module Rm. (2) A module M is (m, n)-injective if and only if M is (n, m)-pure in every module containing M if and only if M is (n, m)-pure in E(M). (3) A module M is (m, n)-flat if and only if M∗ is (m, n)-injective if and only if every exact sequence 0 → U ′ → U → M → 0 is (m, n)-pure. Every (n,ℵ0)-pure submodule of a (m, n)-flat module is (m, n)-flat. (4) If U ′ ≤ U and U is (m, n)-flat, then U/U ′ is (m, n)-flat if and only if U ′ is (m, n)-pure in U . Recall that a ring R is called left IF ring if every injective left R- module is flat [4],[9]. In this paper we shall generalize this notion and introduce left IF-(m, n) rings providing several characterizations of such rings. We also discuss left IF-(1, 1) rings in a number of special cases. Jo u rn al A lg eb ra D is cr et e M at h .Z. Zhanmin, X. Zhangsheng 95 2. Results In this section, m and n will be two fixed positive integers. We start with the following Definition 2.1. A ring R is called left IF-(m, n) ring if every injective left R-module is (m, n)-flat. Theorem 2.2. For any ring R, the following conditions are equivalent: (1) R is a left IF-(m, n) ring. (2) Every FP -injective left R-module is (m, n)-flat. (3) Every (ℵ0, n)-injective left R-module is (m, n)-flat. (4) If RN1 ≤ RN, N1 is (ℵ0, m)-injective and N is (ℵ0, n)-injective, then N/N1 is (m, n)-flat. Proof. (1)⇒(3). If RN is (ℵ0, n) -injective then it is (n,ℵ0)-pure in E(N). Thus, from the (m, n)-flatness of E(N), we have that N is (m, n)- flat. (3)⇒(4). Since N1 is (ℵ0, m)-injective, so N1 is (m,ℵ0)-pure in N and hence (m, n)-pure in N . But N is (m, n)-flat, so N/N1 is (m, n)-flat. The implications (4)⇒(2)⇒(1) are clear. Let P and M be left R-modules. There is a natural homomorphism σ = σP,M : HomR(P, R)⊗M → HomR(P, M) defined by σ(f ⊗m)(p) = f(p)m for f ∈ Hom(P, R), m ∈ M, p ∈ P . Lemma 2.3. Let M be a left R-module. If σP,M is an epimorphism for every (n, m)-presented left R-module P, then M is (m, n)-flat. Proof. For every (n, m)-presented left R-module P and u ∈ HomR(P, M), suppose u = σ( ∑k i=1 fi ⊗ mi), fi ∈ HomR(P, R), mi ∈ M . Let F = Rk, and define v : P → F by v(p) = (f1(p), · · · , fk(p));w : F → M by w(r1, · · · , rk), then u = wv. It follows that P is projective with respect to every exact sequence of left R-modules 0 → M ′ → M ′′ → M → 0. Consequently, M is (m, n)-flat by Theorem 1.2(1),(3). Lemma 2.4. Suppose that RU is (m, n)-flat. The following statements are equivalent (1) RU I is (m, n)-flat for every set I. (2) For every n-generated submodule K of the right R-module Rm, the natural homomorphism ϕK : K ⊗R U I → (K ⊗R U)I defined by ϕK(x ⊗ u)(α) = x ⊗ u(α), α ∈ I, u ∈ U I , x ∈ K, is an isomorphism. Jo u rn al A lg eb ra D is cr et e M at h .96 Rings which have (m, n)-flat injective modules Proof. (1)⇒(2). Assume (1) holds. Then the exact sequence 0 → K → Rm → Rm/K → 0 induces the commutative diagram 0 −−−−→ K ⊗ U I −−−−→ Rm ⊗ U I −−−−→ (Rm/K) ⊗ U I −−−−→ 0   y φK   y φRm   y φRm/K 0 −−−−→ (K ⊗ U)I −−−−→ (Rm ⊗ U)I −−−−→ (Rm/K ⊗ U)I −−−−→ 0 which has exact rows, since RU and RU I are (m, n)-flat. The maps ϕRm and ϕRm/K are isomorphisms since Rm and Rm/K are finitely presented. Hence ϕK is an isomorphism. (2)⇒(1). For every n-generated submodule K of Rm we have the commutative diagram 0 −−−−→ K ⊗ U I −−−−→ Rm ⊗ U I   y φK   y φRm 0 −−−−→ (K ⊗ U)I −−−−→ (Rm ⊗ U)I which has exact bottom row, and ϕK and ϕRm/K are isomorphism by (2). Hence the top row is also exact, and thus U I is (m, n)-flat. Theorem 2.5. For any ring R the following conditions are equivalent (1) R is left IF-(m, n). (2) The injective hull of every finitely presented left R-module is (m, n)-flat. (3) Every (n, m)-presented left R-module is a submodule of a free module. (4) For every free right R-module F , F ∗ is (m, n)-flat. (5) For every n-generated submodule K of the left R-module Rm, the map ϕk : K ⊗R (R∗)I → (K ⊗R R∗)I is an isomorphism. (6) For every m-generated submodule K of the left R-module Rn there exist finite elements β1, β2, · · · , βp in Rn such that K = lRn{β1, β2, · · · , βp}. Proof. (1)⇒(2) is trivial. (2)⇒(3). Let u be the injection of a given (n, m)-presented module RP into E(P ), and let 0 → K → F w −→ E(P ) → 0 be an exact sequence with F being free. Then there exists v : P → F such that u = wv, which implies that v is monic, and (3) holds. (3)⇒(4). Let FR be any free R-module, then F ∗ is injective. Let RP be (n, m)-presented and RL is a finitely generated free module contain- Jo u rn al A lg eb ra D is cr et e M at h .Z. Zhanmin, X. Zhangsheng 97 ing P . The following diagram is commutative Hom(L, R) ⊗ F ∗ σL−−−−→ Hom(L, F ∗)   y α   y β Hom(P, R) ⊗ F ∗ σP−−−−→ Hom(P, F ∗) Since F ∗ is injective, β is an epimorphism. Besides, σL is an isomor- phism since L is finitely generated and free. Therefore σP is epic, and thus F ∗ is (m, n)-flat by Lemma 2.3. (4)⇒(1). Let E be any injective left R-module. There is a free module F and an epimorphism F → E∗, which give a monomorphism E∗∗ → F ∗. Since F ∗ is (m, n)-flat and E ⊆ E∗∗, then E is a direct summand of F ∗ and hence E is (m, n)-flat. (4)⇔(5). Follows from Lemma 2.4. (3)⇒(6). Since K is a m-generated submodule of the left R-module Rn, by (3), Rn/K can be embeded in Rp for some positive integer p. Let f : Rn/K → Rp be a monomorphism and suppose that f(ei) = (ai1, ai2, · · · , aip), where ei is the element in Rn with 1 in the ith position and 0′s in all other positions, i = 1, 2, · · · , n. Write βj = (a1j , a2j , · · · , anj) T , j = 1, 2, · · · , p, then K = lRn{β1, β2, · · · , βp}. (6)⇒(3). Let K = lRn{β1, β2, · · · , βp}, where βj = (a1j , a2j , · · · , anj) T ∈ Rn, j = 1, 2, · · · , p. Write αi = (ai1, ai2, · · · , aip), i = 1, 2, · · · , n, and define f : Rn/K → Rp by (r1, r2, · · · , rn) + K 7−→ (r1, r2, · · · , rn)     a11 a12 · · · a1p a21 a22 · · · a2p · · · · · · · · · · · · an1 an2 · · · anp     then f is a left R-monomorphism. We call a ring R right (m, n)-coherent if every n-generated submodule of Rm R is finitely presented. A straightforward modification of Chase’s proof of [2, Theorem 2.1] shows that R is right (m, n)-coherent if and only if the direct product of any family of (m, n)-flat left R-modules is (m, n)-flat. Corollary 2.6. If R is right (m, n)-coherent then R is left IF-(m, n) if and only if R(R∗) is (m, n)-flat. Proof. Follows from Theorem 2.5(4). Jo u rn al A lg eb ra D is cr et e M at h .98 Rings which have (m, n)-flat injective modules Corollary 2.7. If RR is a cogenerator and R is right (m, n)-coherent then R is left IF-(m, n). Proof. Since R is right (m, n)-coherent, RRI is (m, n)-flat for every set I. Since RR is a cogenerator, every left R-module can be imbedded in an (m, n)-flat module. Consequently, every injective left R-module is a direct summand of an (m, n)-flat module and hence is (m, n)-flat. Corollary 2.8. If R is a left IF -(m, n) ring then R is right (m, n)- injective. Proof. Let K be any m-generated submodule of the left R-module Rn. Since R is left IF -(m, n), K = lRn{β1, β2, · · · , βp} for some β1, β2, · · · , βp in Rn by Theorem 2.5. This implies lRnrRn(K) = K, and thus R is right (m, n)-injective by [3,Theorem 2.4]. Corollary 2.9 [9,Theorem 3.3] A left IF -ring is right FP -injective. According to [4], a right R-module N is called T-finitely generated (resp. H-finitely generated) if N contains a finitely generated submodule N0 such that N/N0 ⊗ R∗ = 0(resp.HomR(N/N0, R) = 0). A right R- module M is called T-finitely presented (resp. H-finitely presented) if there exists a finitely generated free module F and an epimorphism of F onto M with T-finitely generated (resp. H-finitely generated) kernel. We call a ring R right T − (m, n)-coherent (resp. H − (m, n)-coherent) if every n-generated submodule K of Rm R is T -finitely presented (resp. H-finitely presented). Theorem 2.10. Suppose that R(R∗) is flat. The following statements are equivalent (1) R is left IF-(m, n). (2) R is right T-(m, n)-coherent. (3) R is right H-(m, n)-coherent. Proof. (1)⇒(2). Follows from Theorem 2.5(5) and [5,Lemma 1.2]. (2)⇔(3). Follows from [4,Lemma 3,4]. We call a ring R left IF-n ring if every injective left R-module is n-flat (i.e. (1, n)-flat). Jo u rn al A lg eb ra D is cr et e M at h .Z. Zhanmin, X. Zhangsheng 99 Remark 2.11. In general, the classes of left IF -(m, n) rings are dif- ferent for different pairs (m, n). For example, let K be a field and L a proper subfield of K such that ρ : K → L is an isomorphism. Let K[x, ρ] be the ring of twisted left polynomials over K, where xk = ρ(k)x for all k ∈ K. Set R = K[x, ρ]/(x2). It is readily verified that the only proper left ideal of R is Rx = Kx. Thus R is left artinian and so satisfies the ascending chain condition on annihilator right ideals. It is easy to check that Rx = l(x), so R is a left IF -1 ring by Theorem 2.5. As in [11, Example 1], we can show that R is not QF , and thus R is not right 2-injective by [11,Corollary 3]. Therefore, R is not a left IF -2 ring by Corollary 2.8. We characterize left IF-1 rings in some special cases. Theorem 2.12. Let R be a left IF-1 ring. (1) R is a field if and only if R is a domain. (2) R is a regular ring if and only if every principally left ideal is flat. (3) R is a semisimple ring if and only if R is a semiprime left Goldie ring. Proof. We only need to prove the sufficiency. (1) Over a domain, 1-flat modules are torsion free [7, Theorem,3.3]. Thus every injective module, being 1-flat, is torsion free. Hence for every essential ideal I, R/I is torsion free, and so I = R. This means that R is semisimple, and hence is a field. (2) If every principally left ideal is flat then submodules of 1-flat left R- modules are 1-flat by [12, §5, (f)]. We conclude that every left R-module is 1-flat since R is a left IF-1 ring. In particular, R/Ra is 1-flat for each a ∈ R. It follows that Ra is (1, 1)-pure in R, and thus aR ∩ Ra = aRa. Consequently, a ∈ aRa and R is regular. (3) Since R is left IF − 1, then RR is P -injective and hence divisible. This implies that every regular element of R has a left inverse and thus is invertible. Observe that in a semiprime Goldie ring, every essential left ideal has a regular element [8,Theorem 3.9], which is invertible in our case. Therefore R has no proper essential left ideals and hence, R is semisimple. Acknowledgment We are indebted to the referee for several comments that improved the paper. Jo u rn al A lg eb ra D is cr et e M at h .100 Rings which have (m, n)-flat injective modules References [1] F. W. Anderson and K. R. Fuller, Rings and categories of modules, Springer- verlag, 1974,210-211. [2] S. U. Chase, Direct products of modules, Trans.Amer. Math. Soc.97(1960), 457- 473. [3] J. L. Chen, N. Q. Ding, Y. L. Li and Y. Q. Zhou, On (m, n)-injectivity of modules, Comm. Algebra 29(12), 2001, 5589-5603. [4] R. R. Colby, Rings which have flat injective modules, J. Algebra. 35(1975), 239- 252. [5] R. R. Colby and E. A. Rutter, π-flat and π-projective modules, Arch. Math. 22, 1971, 246-251. [6] D. E. Dobbs, On n-flat modules over a Commutative ring, Bull. Austral. Math. Soc. 43, 1991, 491-498. [7] D. E. Dobbs, On the criteria of D. D. Anderson for invertible and flat ideals, Canad. Math. Bull. 29(1), 1986, 26-32. [8] A. W. Goldie, Semi-prime rings with maximum condition, Proc. London Math. Soc. 10(3), 1960. [9] S. Jain, Flat and FP -injectivity, Proc. Amer. Math. Soc. 41(2),1973,437-442. [10] W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra. 174(1995), 77-93. [11] E.A.Rutter, Rings with the principal extension property, Comm. Algebra, 3(3),1975,203-212. [12] A. Shamsuddin, n-injective and n-flat modules, Comm. Algebra. 29(5), 2001, 2039-2050. Contact information Z. Zhanmin, X. Zhangsheng Department of Mathematics, Hubei In- stitute for Nationalities, Enshi, Hubei Province, 445000, P. R. China E-Mail: zhanming_zhu@hotmail.com Received by the editors: 01.07.2004 and final form in 06.05.2005.

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