Category of n-weak injective and n-weak flat modules with respect to special super presented modules
aa r X i v : . [ m a t h . R A ] F e b Category of n-weak injective and n-weak flatmodules with respect to special super presentedmodules
Mostafa Amini ,a , Houda Amzil ,b and Driss Bennis .c
1. Department of Mathematics, Faculty of Sciences, Payame Noor University, Tehran, Iran.2. Department of Mathematics, Faculty of Sciences, Mohammed V University in Rabat, Rabat,Morocco. a . [email protected] b . [email protected]; [email protected] c . [email protected]; driss [email protected] Abstract.
Let R be a ring and n , k two non-negative integers. In this paper, we in-troduce the concepts of n -weak injective and n -weak flat modules and via the notion ofspecial super finitely presented modules, we obtain some characterizations of these mod-ules. We also investigate two classes of modules with richer contents, namely WI nk ( R ) and WF nk ( R op ) which are larger than that of modules with weak injective and weak flatdimensions less than or equal to k . Then on any arbitrary ring, we study the existence of WI nk ( R ) and WF nk ( R op ) covers and preenvelopes. Keywords: n -weak injective module; n -weak flat module; n -super finitely presented; n -weakflat-cover; n -weak flat-envelope. Introduction and Preliminaries
Injectivity of modules is one of the principal notions in homological algebra. Namely, overNoetherian rings, injective modules have very important properties as well as many applicationssince the classical Matlis’s work (see [19]). Stenstr¨om introduced FP-injective modules and stud-ied it over coherent rings as a counterpart notion of injective modules over Noetherian rings (see[29]). Recall that a left R -module M is called FP-injective if Ext R ( U, M ) = 0 for any finitely pre-sented left R -module U . Accordingly, the FP-injective dimension of M , denoted by FP- id R ( M ) ,is defined to be the smallest n ≥ such that Ext n +1 R ( U, M ) = 0 for all finitely presented left R -modules U . If no such n exists, one defines FP- id R ( M ) = ∞ . For background on FP-injective(or absolutely pure) modules, we refer the reader to [9, 18, 20, 21, 22, 25, 24, 29].Recall that coherent rings first appeared in Chase’s paper [4] without being mentioned by name.The term coherent was first used by Bourbaki in [1]. Then, n -coherent rings were introduced byCosta in [6]. Let n be a non-negative integer. A left R -module M is said to be n -presented if thereis an exact sequence F n → F n − → · · · → F → F → M → of left R -modules, where each F i is finitely generated and free. And a ring R is called left n -coherent if every n -presented R -moduleis ( n + 1) -presented. Thus, for n = 1 , left n -coherent rings are nothing but left coherent rings (see[6, 8, 29]).In 2015, Wei and Zhang in [31], introduced the notion of f p n -injective modules as a gener-alization the notion of FP-injective modules by using n -presented modules. They also intro-duce f p n -flat modules. Namely, a left R -module M is said to be f p n -injective if for every ex-act sequence → X → Y with X and Y n -presented left R -modules, the induced sequence Hom(
Y, M ) → Hom(
X, M ) → is exact. And a right R -module N is said to be f p n -flat iffor every exact sequence → X → Y with X and Y n -presented left R -modules, the inducedsequence → N ⊗ R X → N ⊗ R Y is exact. They investigated the properties of these modules andin particular proved the existence of f p n -injective covers (respectively preenvelopes) and f p n -flatcovers (respectively preenvelopes) (see [31, Theorem 2.5]). On the other hand, Bravo et al. in [2]introduced the notion of absolutely clean and level modules, which Gao and Wang in [11] renamedby weak injective and weak flat, respectively.In this paper, we deal with weak injective and weak flat modules and some extensions of thesenotions. In 2015, Gao and Wang in [11], using super finitely presented modules instead of finitelypresented modules, introduced the concept of weak injective and weak flat modules. In general, eak injective and weak flat modules are generalization of f p n -injective and f p n -flat modules,respectively. A left R -module U is called super finitely presented if there exists an exact sequence · · · −→ F −→ F −→ F −→ U −→ , where each F i is finitely generated and free. A left R -module M is called weak injective if Ext R ( U, M ) = 0 for any super finitely presented U . Aright R -module M is called weak flat if Tor R ( M, U ) = 0 for any super finitely presented U .Using weak injective and weak flat modules, several authors investigated the homological as-pect of some notions on arbitrary rings, and concluded the results of some concepts already doneon coherent rings. For example, in 2018, Zhao in [34] investigated the homological aspect ofmodules with finite weak injective and weak flat dimensions. For instance; if WI k ( R ) and WF k ( R op ) are classes of left and right modules with weak injective dimension and weak flat di-mensions less than or equal to k , respectively, then by using derived functors Ext WF , Ext WI and Tor W on WF ( R op ) -resolutions and WI ( R ) -resolutions, it is proved that the classes WI k ( R ) and WF k ( R op ) are covering and preenveloping over any arbitrary ring, where WI ( R ) and WF ( R op ) are the class of weak injective left modules and weak flat right modules, respectively. And when k = 0 and for coherent rings, the result is that every module has an FP-injective cover and anFP-injective preenvelope. In the recent years, the homological theory for weak injective and weakflat modules have become an important area of research (see [11, 12, 35]).We let n, k be non-negative integers. In this paper, we introduce a concept of n -weak injec-tive left modules and n -weak flat right modules by using n -super finitely presented left modules.Every n -weak injective (resp. n -weak flat) is weak injective (resp. weak flat). And if n ≥ ,then n -weak injective (resp. n -weak flat) modules are common generalization of weak injectiveand f p n -injective (resp. weak flat and f p n -flat) modules. Under these definitions, n -super finitelypresented, n -weak injective and n -weak flat modules are weaker than the usual super finitely pre-sented, weak injective (resp. f p n -injective) and weak flat (resp. f p n -flat) modules, respectively.Also, for any m ≥ n , every n -super finitely presented, n -weak injective (resp. f p m -injective) and n -weak flat (resp. f p m -flat) modules is n -super finitely presented, n -weak injective and n -weakflat, respectively. But, n -weak injective and n -weak flat R-modules need not be m -weak injective(resp. f p m -injective) and m -weak flat (resp. f p m -flat) for any m < n (resp. m ≤ n ), (see Ex-ample 2.5). We also introduce and investigate the classes WI nk ( R ) and WF nk ( R op ) larger than theclasses WI k ( R ) and WF k ( R op ) (resp. f p n I and f p n F ) in [34] (resp. [31]).The paper is organized as follows: n Sec. 1, some fundamental concepts and some preliminary results are stated.In Sec. 2, we introduce n -super finitely presented, n -weak injective left modules and n -weak flatright modules. Then, by considering special super finitey presented R -modules of every n -superfinitely presented left R -module, we give some characterizations of these modules.In Sec. 3, we give some homological aspects of modules with finite n -weak injective and n -weakflat dimensions. We let WI nk ( R ) and WF nk ( R op ) denote the classes of left and right moduleswith n -weak injective dimension and n -weak flat dimensions less than or equal to k , respectively.Among other results, we prove that on any arbitrary ring , M is in WF nk ( R op ) (resp. WI nk ( R ) ) ifand only if M ∗ is in WI nk ( R ) (resp. WF nk ( R op ) ), where M ∗ = Hom Z ( M, QZ ) . Also, consideringthe exact sequence → A → B → C → of R -modules, we show that C is in WI nk ( R ) if A and B are in WI nk ( R ) and A is in WF nk ( R op ) if B and C are in WF nk ( R op ) . Then, from this resultin Section 4 and over any arbitrary ring, we show that WI nk ( R ) and WF nk ( R op ) are injectivelyresolving and projectively resolving and consequently, we prove that the classes WI nk ( R ) and WF nk ( R op ) are covering and preenveloping. Then by considering n = 0 , we deduce that theclasses WI k ( R ) and WF k ( R op ) are covering and preenveloping ([34, Theorems 4.4, 4.5, 4.8 and4.9]). Moreover, if k = 0 , then Theorems 2.5 and 2.7 of [31] follow. Also, we investigate ringsover which every left module is in WI n ( R ) and every right module is in WF n ( R op ) . Finally, weshow that the pair ( WF nk ( R op ) , WF nk ( R op ) ⊥ ) is a hereditary perfect cotorsion pair, and if R is in WI nk ( R ) , it follows that the pair ( WI nk ( R ) , WI nk ( R ) ⊥ ) is perfect cotorsion pair. n -Weak injective and n -weak flat modules In this section, we introduce the notions of n -weak injective and n -weak flat modules usingspecial super finitely presented modules. Then, we show some of their general properties. We startwith the following definition. Definition 2.1.
Let n be a non-negative integer. A left R -module U is said to be n -super finitelypresented if there exists an exact sequence · · · −→ F n +1 −→ F n −→ · · · −→ F −→ F −→ U −→ of projective R -modules, where each F i is finitely generated projective for any i ≥ n . f K i := Im( F i +1 → F i ) , then for i = n − , we call the object K n − special super finitelypresented. Moreover, if Hom R ( K n − , − ) is exact with respect to the short exact sequence → A → B → C → of left R -modules, then we say that this sequence is special superpure and A issaid to be superpure in B . Definition 2.2.
Let n be a non-negative integer. A left R -module M is called n -weak injective if Ext n +1 R ( U, M ) = 0 for every n -super finitely presented left R -module U . A right R -module N iscalled n -weak flat if Tor Rn +1 ( N, U ) = 0 for every n -super finitely presented left R -module U . Remark 2.3.
Let n, m, k be non-negative integers. Then: (1)
Ext n +1 R ( U, − ) ∼ = Ext R ( K n − , − ) and Tor Rn +1 ( − , U ) ∼ = Tor R ( − , K n − ) , where U is an n -super finitely presented with special super finitely presented K n − . In case n = 0 , then n -weak injective left R -modules, n -super finitely presented left R -modules and n -weak flatright R -modules are simply weak injective left R -modules, super finitely presented left R -module and weak flat right R -module, respectively. (2) Every super finitely presented left R -module is n -super finitely presented. (3) Every n -super finitely presented left R -module is m -super finitely presented for any m ≥ n ,but not conversely (see Examples 2.4 and 2.5(1)). If we denote by Pres ∞ n the class of all n -super finitely presented left R -modules, then: Pres ∞ n ⊆ Pres ∞ n +1 ⊆ Pres ∞ n +2 ⊆ · · · If n = 0 , then Pres ∞ is simply the class of all super finitely presented left R -modules. Wedenote this class simply by Pres ∞ . (4) Every n -weak injective left (resp. n -weak flat right) R -module is m -weak injective (resp. m -weak flat) for any n ≤ m , but not conversely (see Example 2.5(2)). If U is an ( n + 1) -superfinitely presented left R -module, then there exists the following exact sequence: · · · −→ F −→ F −→ F −→ U −→ , where K n is special super finitely presented and also, the short exact sequence → K → F → U → exists, where K is n -super finitely presented. So, if M is a n -weak in-jective left R -module, then Ext n +1 R ( K , M ) = 0 . On the other hand, Ext n +2 R ( U, M ) ∼ = xt n +1 R ( K , M ) = 0 , and hence M is ( n + 1) -weak injective. Similarly, every n -weak flatright R -module is ( n + 1) -weak flat. (5) If I , F P , WI ( R ) , WI n ( R ) , F , WF ( R op ) and WF n ( R op ) are the classes of injective,FP-injective, weak injective, n -weak injective left R -modules and flat, weak flat and n -weakflat right R -modules, respectively. Then, I ⊆ F P ⊆ WI ( R ) ⊆ WI n ( R ) ⊆ WI n +1 ( R ) ⊆ WI n +2 ( R ) ⊆ · · · and F ⊆ WF ( R op ) ⊆ WF n ( R op ) ⊆ WF n +1 ( R op ) ⊆ WF n +2 ( R op ) ⊆ · · · . (6) Every f p n -injective left R -module is n -weak injective and every f p n -flat right R -moduleis n -weak flat. Indeed, for any n -super finitely presented left module U , there exists theshort exact sequence → K n → F n → K n − → , where K n − is special super finitelypresented. So, if M is f p n -injective left R -module, then Hom( F n , M ) → Hom( K n , M ) → is exact, since K n and F n are super finitely presented and consequently are n -presented.Therefore, Ext R ( K n − , M ) = 0 and thus by (1), it follows that M is n -weak injective.Similarly, every f p n -flat right R -module is n -weak flat, but not conversely (see Example2.5(3)). Let R be a ring and A an R -module. Then, the finitely presented dimension of A denoted by f . p . dim R ( A ) is defined as inf { n | there exists an exact sequence F n +1 → F n → · · · → F → F → A → R - modules , where each F i projective , and F n and F n+1 are finitely generated } .So, f . p . dim( R ) = sup { f . p . dim R ( A ) | A is a finitely generated R - module } . We use w . gl . dim R and gl . dim R to denote the weak global dimension and global dimension respectively. Also, a ring R is called ( a, b, c ) -ring, if w . gl . dim R ( R ) = a , gl . dim R ( R ) = b and f . p . dim R ( R ) = c (see [23]). Example 2.4.
Let R = k [ x , x ] ⊕ R ′ , where k [ x , x ] is a ring of polynomials in indeterminateover a field k , and R ′ is a valuation ring with global dimension . Then by [23, Proposition 3.10], R is a coherent (2 , , -ring. So f . p . dim R ( R ) = 3 , and hence there exists a finitely generated R -module U such that f . p . dim R ( U ) = 3 . Thus, there exists an exact sequence F → F → F → F → F → U → , where F and F are finitely generated projective modules. Also, K := Im( F → F ) is special super finitely presented, since R is coherent. So by Definition 2.1, U is -super finitely presented. But, U is not -super finitely presented otherwise f . p . dim R ( U ) = 2 ,a contradiction. xample 2.5. Let R = k [ x , x ] ⊕ S , where k [ x , x ] is a ring of polynomials in indeterminateover a field k , and S is a non-Noetherian hereditary von Neumann regular ring (for example; S isa ring of functions of X into k continuous with respect to the discrete topology on k , where k isa field and X is a totally disconnected compact Hausdorff space whose associated Boolean ringis hereditary, see examples of [3]). Then by [23, Proposition 3.8], R is a coherent (2 , , -ring.Hence, (1) Since f . p . dim R ( R ) = 2 , then by [23, Proposition 1.5], for a finitely generated R -module U either f . p . dim R ( U ) = 2 or f . p . dim R ( U ) = 0 . If f . p . dim R ( U ) = 0 , it follows that U is finitely presented. If f . p . dim R ( U ) = 2 , then there exists an exact sequence F → F → F → F → U → , where F and F are finitely generated projective. Also, K :=Im( F → F ) is special super finitely presented, since R is coherent. So by Definition2.1, U is -super finitely presented. But, U is neither -super finitely presented nor -superfinitely presented, otherwise every finitely generated R -module would be finitely presented,and hence by [23, Theorem 1.3], f . p . dim R ( R ) = 0 , a contradiction. (2) It is clear that every R -module is -weak injective, since gl . dim R ( R ) = 2 . But not ev-ery R -module is -weak injective. Indeed, if every module M is -weak injective, then Ext R ( U ′ , M ) = 0 for each -super finitely presented U ′ . So, each -super finitely pre-sented is projective. But since R is coherent, any finitely presented module is super finitelypresented, and so any finitely presented module is projective. Hence R is -regular, and thenby [33, Theorem 3.9], every R -module is flat. So, w . gl . dim R ( R ) = 0 , a contradiction. Sim-ilarly, since w . gl . dim R ( R ) = 2 , it follows that every R -module -weak flat, but not every R -module -weak flat. (3) Not every R -module is f . p -injective (resp. f . p -flat). Indeed, suppose otherwise. Since R is coherent, it follows that any finitely presented left R -module A is super finitely presented.Then, there is an exact sequence → L → F → A → , where F and L are n -presented, and so are also -presented. So, if M is a f . p -injective, then Ext R ( A, M ) = 0 (resp.
Tor R ( M, A ) = 0 ) and consequently, A is projective. Hence, R is -regular and thus w . gl . dim R ( R ) = 0 , a contradiction. We denote by R - Mod the category of left R -modules and by R - Mod that of right R -modules. Proposition 2.6.
Let n be a non-negative integer. Then, For every M ∈ Mod - R , M is n -weak flat if and only if M ∗ is n -weak injective. (2) For every M ∈ R - Mod , M is n -weak injective if and only if M ∗ is n -weak flat.Proof. (1) Let U be an n -super finitely presented left R -module with special super finitely pre-sented K n − . Then, Tor R ( M, K n − ) ∗ ∼ = Ext R ( K n − , M ∗ ) , since by [26, Theorem 2.76], Hom Z ( M ⊗ R K n − , QZ ) ∼ = Hom R ( K n − , Hom Z ( M, QZ )) . The result follows then using Remark 2.3(1), since Ext n +1 R ( U, M ∗ ) ∼ = Ext R ( K n − , M ∗ ) ∼ = Tor R ( M, K n − ) ∗ ∼ = Tor Rn +1 ( M, U ) ∗ .(2) Let U be an n -super finitely presented left R -module with special super finitely presented K n − . Then, Ext R ( K n − , M ) ∗ ∼ = Tor R ( M ∗ , K n − ) , since by [26, Lemma 3.55], we havethat M ∗ ⊗ R K n − ∼ = Hom R ( K n − , M ) ∗ . The result follows then using Remark 2.3(1), since Tor Rn +1 ( M ∗ , U ) ∼ = Tor R ( M ∗ , K n − ) ∼ = Ext R ( K n − , M ) ∗ ∼ = Ext n +1 R ( U, M ) ∗ . As a direct consequence of Proposition 2.6 we obtain the following corollary
Corollary 2.7.
Let M be an R -module. Then, (1) For every M ∈ R - Mod , M is n -weak injective if and only if M ∗∗ is n -weak injective. (2) For every M ∈ Mod - R , M is n -weak flat if and only if M ∗∗ is n -weak flat. Proposition 2.8.
Let M be a left R -module. Then, the following statements are equivalent: (1) M is n -weak injective; (2) Every short exact sequence → M → B → C → of left R -modules is special superpure; (3) M is special superpure in any left R -module containing it; (4) M is special superpure in any injective left R -module containing it; (5) M is special superpure in E ( M ) .Proof. (1) = ⇒ (2) Let U be an n -super finitely presented left R -module with special super finitelypresented K n − . Then by Remark 2.3(1), Ext n +1 R ( U, M ) ∼ = Ext R ( K n − , M ) = 0 . Consequently, Hom R ( K n − , − ) is exact with respect to the short exact sequence → M → B → C → . (2) = ⇒ (3) = ⇒ (4) = ⇒ (5) are clear.
5) = ⇒ (1) The short exact sequence → M → E ( M ) → E ( M ) M → is special superpure.Therefore, if U is an n -super finitely presented left R -module with special super finitely presented K n − , then by assumption and Remark 2.3(1), R ( K n − , M ) ∼ = Ext n +1 R ( U, M ) and hence M is n -weak injective. Proposition 2.9.
Let n be a non-negative integer. Then, the following assertions hold: (1) Let { M i } i ∈ I be a family of left R -modules. Then Q i ∈ I M i is n -weak injective if and only ifeach M i is n -weak injective. (2) Let { M i } i ∈ I be a family of right R -modules. Then L i ∈ I M i is n -weak flat if and only ifeach M i is n -weak flat.Proof. Both assertions follow easily using [26, Proposions 7.6 and 7.22].
Proposition 2.10.
Let n be a non-negative integer and let { M i } i ∈ I be a family of left R -modules.Then L i ∈ I M i is n -weak injective if and only if each M i is n -weak injective.Proof. If U is an n -super finitely presented left R -module with special super finitely presented K n − , then there exists the short exact sequence → K n → F n → K n − → , where F n is finitely presented projective and K n is super finitely presented. Thus we have the followingcommutative diagram: Hom R ( F n , L i ∈ I M i ) / / ∼ = (cid:15) (cid:15) Hom R ( K n , L i ∈ I M i ) / / ∼ = (cid:15) (cid:15) Ext R ( K n − , L i ∈ I M i ) / / (cid:15) (cid:15) L i ∈ I Hom R ( F n , M i ) / / L i ∈ I Hom R ( K n , M i ) / / L i ∈ I Ext R ( K n − , M i ) / / So,
Ext R ( K n − , L i ∈ I M i ) ∼ = L i ∈ I Ext R ( K n − , M i ) and hence by Remark 2.3(1), L i ∈ I M i is n -weak injective if and only if every M i is n -weak injective. Proposition 2.11.
Let n be a non-negative integer. Then, the following assertions hold:1. A left R -module M is n -weak injective if and only if every pure submodule and pure epi-morphic image of M is n -weak injective. . A right R -module M is n -weak flat if and only if every pure submodule and pure epimorphicimage of M is n -weak flat.Proof. (1) Let M be an n -weak injective left R -module and N a pure submodule of M . Then thereexists a pure exact sequence → N → M → MN → which gives rise to a split exact sequence → ( MN ) ∗ → M ∗ → N ∗ → . By Proposition 2.6(2), M ∗ is n -weak flat. Now by Proposition2.9(2), M ∗ is n -weak flat if and only if N ∗ and ( MN ) ∗ are n -weak flat. Hence by Proposition 2.6(2),we deduce that M is n -weak injective if and only if N and MN are n -weak injective. Similarly, weprove (2) using Propositions 2.6(1) and 2.9(1).Now, from the previous results in this section, the following results are obtained. Theorem 2.12.
Any direct product of n -weak flat right R -modules is n -weak flat.Proof. Let { M i } i ∈ I be a family of n -weak flat right R -modules. By [32, Proposition 2.4.5], M i is pure in M ∗∗ i , hence Q i ∈ I M i is pure in Q i ∈ I M ∗∗ i . Thus, using Corollary 2.11(2), it suffices toprove that Q i ∈ I M ∗∗ i is n -weak flat. Using [27, Theorem 2.4], we have Q i ∈ I M ∗∗ i ∼ = ( L M ∗ i ) ∗ .By Proposition 2.6(1), each M ∗ i is n -weak injective , and so Q i ∈ I M ∗ i is n -weak injective byProposition 2.9(1). Thus ( L M ∗ i ) ∗ is n -weak flat by Proposition 2.9(2). Proposition 2.13.
Let R be a ring. Then (1) If a left R -module M is n -weak injective, then Ext iR ( K n − , M ) = 0 for every special superfinitely presented K n − and i ≥ . (2) If a right R -module M is n -weak flat, then Tor Ri ( M, K n − ) = 0 for every special superfinitely presented K n − and i ≥ .Proof. (1) Let U be an n -super finitely presented left R -module with special super finitely pre-sented K n − . Then by Remark 2.3(3), U is ( n + 1) -super presented with special super finitelypresented K n . Also by Remark 2.3(4), every n -weak injective is ( n + 1) -weak injective, and hence Ext n +2 R ( U, M ) ∼ = Ext R ( K n , M ) = 0 . There is an exact sequence → K n → F n → K n − → ,where F n is finitely generated and projective. It follows that Ext R ( K n , M ) ∼ = Ext R ( K n − , M ) and so Ext R ( K n − , M ) = 0 . Repeating this process, we conclude that Ext iR ( K n − , M ) = 0 forevery i ≥ .
2) Assume that U is an n -super finitely presented left R -module with special super finitely pre-sented K n − . By Remark 2.3(4), every n -weak flat is ( n +1) -weak flat, and hence Tor Rn +2 ( M, U ) ∼ =Tor R ( M, K n ) = 0 . We have the following short exact sequence → K n → F n → K n − → where F n is finitely generated and projective. We deduce that Tor R ( M, K n ) ∼ = Tor R ( M, K n − ) ,then Tor R ( M, K n − ) = 0 , and so, Tor Ri ( M, K n − ) = 0 for any i ≥ . Corollary 2.14. ([11, Propositin 3.1])(1)
If a left R -module M is weak injective, then Ext iR ( F, M ) = 0 for every super finitelypresented F and i ≥ . (2) If a right R -module M is weak flat, then Tor Ri ( M, F ) = 0 for every super finitely presented F and i ≥ . In this section, we give some homological aspects of modules with finite n -weak injective and n -weak flat dimensions. Let n , k be non-negative integers. We denote by WI nk ( R ) and WF nk ( R op ) the classes of left and right modules with n -weak injective dimension and n -weak flat dimensionsless than or equal to k , respectively. If k = 0 , then WI n ( R ) = WI n ( R ) and WF n ( R op ) = WF n ( R op ) , see Remark 2.3(5). Definition 3.1.
Let R be a ring. Then, the n -weak injective dimension of a left module M and n -weak flat dimension of a right module N are defined by n - wid R ( M ) = inf { k : Ext k +1 R ( K n − , M ) = 0 f or every special super f initely presented K n − } ,and n - wfd R ( N ) = inf { k : Tor Rk +1 ( N, K n − ) = 0 f or every special super f initely presented K n − } . If k = 0 , then by Remark 2.3(1), it follows that M and N are n -weak injective and n -weak flat,respectively. Proposition 3.2.
Let R be a ring and M a left R -module. Then, the following statements areequivalent: n - wid R ( M ) ≤ k ; (2) Ext k + iR ( K n − , M ) = 0 for any special super finitely presented left R -module K n − andany i ≥ ; (3) Ext k +1 R ( K n − , M ) = 0 for any special super finitely presented left R -module K n − ; (4) If the sequence → M → E → E → · · · → E k → is exact with E , E , · · · E k − n -weak injective, then E k is also n -weak injective; (5) There exists an exact sequence → M → E → E → · · · → E k → , where each E i is n -weak injective.Proof. (2) = ⇒ (3) = ⇒ (1) and (4) = ⇒ (5) are obvious. (1) = ⇒ (2) We proceed by induction on k . If k = 0 , then the result follows by Theorem 2.13(1).Suppose now the following assertion: if n - wid R ( M ) ≤ k , then Ext k + iR ( K n − , M ) = 0 for anyspecial super finitely presented left R -module K n − and any i ≥ .Let us prove the following assertion: if n - wid R ( M ) ≤ k + 1 , then Ext k +1+ iR ( K n − , M ) = 0 forany special super finitely presented left R -module K n − and any i ≥ .Suppose that n - wid R ( M ) ≤ k + 1 . If n - wid R ( M ) ≤ k ; then by induction hypothesis we have Ext k + iR ( K n − , M ) = 0 for any special super finitely presented left R -module K n − and any i ≥ ,and so Ext k +1+ iR ( K n − , M ) = 0 for any special super finitely presented left R -module K n − andany i ≥ . Now, if n - wid R ( M ) = k + 1 , then Ext k +2 R ( K n − , M ) = 0 for any special super finitelypresented left R -module.By induction on i , we prove that Ext k +1+ iR ( K n − , M ) = 0 for any special super finitely pre-sented left R -module K n − and i ≥ . The assertion is clearly true for i = 1 . Suppose nowthat Ext k +1+ iR ( K n − , M ) = 0 for any special super finitely presented left R -module K n − andlet us prove that Ext k +2+ iR ( K n − , M ) = 0 for any special super finitely presented left R -module K n − . For let U be an n -super presented left R -module with special super finitely presented K n − . Consider the following short exact sequence → K n → F n → K n − → . Then Ext k + i +1 R ( K n , M ) ∼ = Ext k + i +2 R ( K n − , M ) . But K is n -super finitely presented with special su-per finitely presented K n , hence by induction hypothesis Ext k + i +1 R ( K n , M ) = 0 and consequently Ext k + i +2 R ( K n − , M ) = 0 which completes the proof.
2) = ⇒ (4) Consider the short exact sequence → M → E → E /M → . Then for anyspecial super finitely presented left R -module K n − , we have · · · → Ext k +1 R ( K n − , E ) → Ext k +1 R ( K n − , E /M ) → Ext k +2 R ( K n − , M ) → · · · But by assumption
Ext k +2 R ( K n − , M ) = 0 and by Proposition 2.13(1) Ext k +1 R ( K n − , E ) , so Ext k +1 R ( K n − , E /M ) = 0 . Step by step, we prove that Ext R ( K n − , E k ) = 0(5) = ⇒ (1) follows from Proposition 2.13(1). Proposition 3.3.
Let R be a ring and M an right R -module. Then, the following statements areequivalent: (1) n - wfd R ( M ) ≤ k ; (2) Tor Rk + i ( M, K n − ) = 0 for any special super finitely presented left R -module K n − and any i ≥ ; (3) Tor Rk +1 ( M, K n − ) = 0 for any special super finitely presented left R -module K n − ; (4) If the sequence → F k → F k − → · · · → F → F → is exact with F , F , · · · F k − n -weak injective, then also F k is n -weak flat; (5) There exists an exact sequence → F k → F k − → · · · → F → F → , where each F i is n -weak flat.Proof. The proof is similar to the proof of Proposition 3.2, using Proposition 2.13(2).
Corollary 3.4.
Let → A → B → C → be a short exact sequence of R -modules. Then (1) C is in WI nk ( R ) if A and B are in WI nk ( R ) . (2) A is in WF nk ( R op ) if B and C are in WF nk ( R op ) .Proof. (1) Let U be an n -super finitely presented left R -module with special super finitely pre-sented K n − . If A and B are in WI nk ( R ) , then by Proposition 3.2, the short exact sequence → A → B → C → induces the following exact sequence: k +1 R ( K n − , B ) −→ Ext k +1 R ( K n − , C ) −→ Ext k +2 R ( K n − , A ) = 0 . ence Ext k +1 R ( K n − , C ) = 0 , and so C is in WI nk ( R ) .(2) Assume that → A → B → C → is a short exact sequence of right R -modules. Then, byhypothesis and Proposition 3.3, We have: Rk +2 ( K n − , C ) −→ Tor Rk +1 ( K n − , A ) −→ Tor Rk +1 ( K n − , B ) = 0 , where K n − is special super finitely presented of n -super presented left R -module U . Conse-quently, Tor Rk +1 ( K n − , A ) = 0 and so, A is in WF nk ( R op ) .The following proposition is a special case of [11, Proposition 3.5]. Proposition 3.5.
Let R be a ring and M an R -module. Then, the following statements are equiv-alent: (1) pd R ( K n − ) ≤ k for all special super finitely presented left R -modules K n − . (2) fd R ( K n − ) ≤ k for all special super finitely presented left R -modules K n − . (3) Ext k +1 R ( K n − , M ) = 0 for any special super finitely presented left R -modules K n − andany left R -module M . (4) Tor Rk +1 ( M, K n − ) = 0 for any special super finitely presented left R -modules K n − andany right R -module M . Proposition 3.6.
Let R be a ring and M an R -module. Then (1) M is in WF nk ( R op ) if and only if M ∗ is in WI nk ( R ) . (2) M is in WI n k ( R ) if and only if M ∗ is in WF nk ( R op ) .Proof. (1) We proceed by induction on k . If k = 0 , then by Propositions 2.6(1) and 3.3, M is n -weak flat if and only if M ∗ is n -weak injective. Consider, the short exact sequence → M → E → L → , where E is injective. Then, M is in WF nk ( R op ) if and only if L is in WF nk − ( R op ) .We have L is in WF nk − ( R op ) if and only if L ∗ is in WI nk − ( R ) and so M ∗ is in WI nk ( R ) .Similarly, by Propositions 2.6(2) and 3.2, (2) is holds. Proposition 3.7.
Let M be an R -module. Then (1) M is in WI nk ( R ) if and only if for every pure submodule N of M , N and MN are in WI nk ( R ) . M is in WF nk ( R op ) if and only if for every pure submodule N of M , N and MN are in WF nk ( R op ) .Proof. Similar to proof of the Proposision 2.11 with by using Proposition 3.6. n -weak in-jective and n -weak flat dimension at most k For a ring R , let Y be a class of R -modules and M be an R -module. Following [9], we say that amorphism f : F → M is a Y -precover of M if F ∈ Y and Hom R ( F ′ , F ) → Hom R ( F ′ , M ) → is exact for all F ′ ∈ Y . Moreover, if whenever a morphism g : F → F such that f g = f isan automorphism of F , then f : F → M is called an Y -cover of M . The class Y is called(pre)covering if each object in R has a Y -(pre)cover. Dually, the notions of Y -preenvelopes, Y -envelopes and (pre)enveloping classes are defined.A duality pair over R [16] is a pair ( M , C ) , where M is a class of left R -modules and C is aclass of right R - modules, subject to the following conditions: (1) For an R -module M , one has M ∈ M if and only if M ∗ ∈ C . (2) C is closed under direct summands and finite direct sums.A duality pair ( M , C ) is called (co)product-closed if the class M is closed under (co)products inthe category of all left R -modules. A duality pair ( M , C ) is called perfect if it is coproduct-closed,if M is closed under extensions, and if R belongs to M .Let X be a class of R -modules. We denote by I ( R ) the class of finite injective left modulesand by F ( R ) the class of finite projective right modules. We call X injectively resolving if I ( R ) ⊆ X , and for every short exact sequence → A → B → C → with A ∈ X the conditions B ∈ X and C ∈ X are equivalent. Also, we call X projectively resolving if F ( R ) ⊆ X , and for every short exact sequence → A → B → C → with C ∈ X theconditions A ∈ X and B ∈ X are equivalent, see [15, 1.1. Resolving classes].In this section, by the use of duality pairs, we investigate WI nk ( R ) and WF nk ( R op ) as preen-veloping and covering classes Proposition 4.1.
The pair ( WI nk ( R ) , WF nk ( R op )) is a duality pair.Proof. Let { M i } i ∈ I be a family of right R -modules. If every M i is in WF nk ( R op ) , then we claimthat L i ∈ I M i is in WF nk ( R op ) . By induction, if k = 0 , then by Proposition 2.9(2), L i ∈ I M i is -weak flat. For every R -module M i , there exists the short exact sequence → L i → P i → M i → of right R -modules, where P i is projective. Thus we have the following short exactsequence → L i ∈ I L i → L i ∈ I P i → L i ∈ I M i → . Since each L i is in WF nk − ( R op ) , by theinduction hypothesis, we have that L i ∈ I L i is in WF nk − ( R op ) , and so it follows that L i ∈ I M i isin WF nk ( R op ) . Also by Proposition 3.6(2), M is in WI nk ( R ) if and only if M ∗ is in WF nk ( R op ) .On the other hand, by Corollary 3.4(2), the class WF nk ( R op ) is projectively resolving, and then by[15, Proposition 1.4], the class WF nk ( R op ) is closed under direct summands, and so we concludethat pair ( WI nk ( R ) , WF nk ( R op )) is a duality pair. Proposition 4.2.
The pair ( WF nk ( R op ) , WI nk ( R )) is a duality pair.Proof. By Proposition 3.6(1), M is in WF nk ( R op ) if and only if M ∗ is in WI nk ( R ) . Let { M i } i ∈ I bea family of left R -modules. Suppose that each M i is in WI nk ( R ) and let us show that Q i ∈ I M i is in WI nk ( R ) . By induction, if k = 0 , then by Proposition 2.9(1), Q i ∈ I M i is n -weak injective. Thereexists the short exact sequence → M i → E i → D i → of left R -modules, where E i is injective,and consequently, we have the following short exact sequence → Q i ∈ I M i → Q i ∈ I E i → Q i ∈ I D i → . Since each D i is in WI nk − ( R ) , by induction hypothesis, we deduce that Q i ∈ I D i is in WI nk − ( R ) , and it follows easily that Q i ∈ I M i is in WI nk ( R ) . So in particular, every finitedirect sum of family { M i } i ∈ I in WI nk ( R ) is in WI nk ( R ) . Also by Corollary 3.4(1), it is clearthat the class WI nk ( R ) is injectively resolving, and so by [15, Proposition 1.4], the class WI nk ( R ) is closed under direct summands, and hence, it follows that the pair ( WF nk ( R op ) , WI nk ( R )) is aduality pair. Lemma 4.3. ([16, Theorem 3.1])
Let ( M , C ) is a duality pair. Then M is closed under puresubmodules, pure quotients, and pure extensions. Furthermore, the following hold: (1) If ( M , C ) is product-closed then M is preenveloping. (2) If ( M , C ) is coproduct-closed then M is covering. (3) If ( M , C ) is perfect then ( M , M ⊥ ) is a perfect cotorsion pair. Theorem 4.4.
The class WI nk ( R ) is covering and preenveloping.Proof. By Proposition 3.7(1), the class WI nk ( R ) is closed under pure submodules, pure quotients,and pure extensions. Also, by the proof of the Proposition 4.2, the class WI nk ( R ) is closed under irect product, and similarly by using Proposition 2.10, we see that the class WI nk ( R ) is also closedunder direct sums. By Proposition 4.1, the pair ( WI nk ( R ) , WF nk ( R op )) is a duality pair, and sofrom Lemma 4.3 follows that the class WI nk ( R ) is covering and preenveloping. Theorem 4.5.
The class WF nk ( R op ) is covering and preenveloping.Proof. By Proposition 3.7(2), the class WF nk ( R op ) is closed under pure submodules, pure quo-tients, and pure extensions. Now, we show that the class WF nk ( R op ) is closed under direct product.If { M i } i ∈ I is a family of right R -modules, where M i is in WF nk ( R op ) for any i ∈ I , then by induc-tion, if k = 0 , by Theorem 2.12, it follows that M i is n -weak flat if and only if Q i ∈ I M i is n -weakflat. Consider, the short exact sequence → L i → P i → M i → of right R -modules, where P i isprojective. Then, there exists the short exact sequence → Q i ∈ I L i → Q i ∈ I P i → Q i ∈ I M i → .If M i is in WF nk ( R op ) , then L i is in WF nk − ( R op ) . By induction hypothesis, Q i ∈ I L i is in WF nk − ( R op ) , and so, we conclude that Q i ∈ I M i is in WF nk ( R op ) . Similarly, using Proposition2.9(2), we see that the class WF nk ( R op ) is closed under direct sums. Since the pair ( WF nk , WI nk ) is a duality pair by Proposition 4.2, we conclude that the class WF nk ( R op ) is covering and preen-veloping from Lemma 4.3.By considering the classes of modules WI nk ( R ) and WF nk ( R op ) , if n = 0 , then WI k ( R ) = WI k ( R ) and WF k ( R op ) = WF k ( R op ) . Since if n = 0 , then by Remark 2.3(1), WI k ( R ) and WF k ( R op ) the classes of left R -modules and right R -modules with weak injective dimension andweak flat dimension less than or equal to k , respectively, see [34]. Thus by Proposition 3.6 andTheorems 4.4 and 4.5, we have the following result: Corollary 4.6. ([34, Theorems 4.4, 4.5, 4.8 and 4.9])(1)
The class WI k ( R ) is covering and preenveloping. (2) The class WF k ( R op ) is covering and preenveloping. Also, if k = 0 , then of Proposition 3.6 and Theorems 4.4 and 4.5 we have: Corollary 4.7. ([31, Theorem 2.5])
For any ring R , the following claims are true: (1) Every left (resp. right) R -module has an f p n -injective (resp. f p n -flat) cover. (2) Every left (resp. right) R -module has an f p n -injective (resp. f p n -flat) preenvelope. If M → N is an f p n -injective (resp. f p n -flat) preenvelope of a left (resp. right) R -module M , then N ∗ → M ∗ is an f p n -flat (resp. f p n -injective) precover of M ∗ . Now we give some equivalent results for R R ∈ WI nk ( R ) in terms of the properties of WI nk ( R ) and WF nk ( R op ) . Proposition 4.8.
Let R be a ring. Then, the following statements are equivalent: (1) R R is in WI nk ( R ) ; (2) Every right R -module has a monic WF nk ( R op ) -preenvelope; (3) Every injective right R -module is in WF nk ( R op ) ; (4) Every flat left R -module is in WI nk ( R ) ; (5) Every projective left R -module is in WI nk ( R ) ; (6) Every left R -module has an epic WI nk ( R ) -cover.Proof. (1) = ⇒ (2) By Theorem 4.5, every R -module M has a WF nk ( R op ) -preenvelope f : M → F . By Proposition 3.6(2), R ∗ is in WF nk ( R op ) , and so similar to proof the of Theorem 4.5, Q i ∈ I R ∗ is in WF nk ( R op ) . Also, ( R R ) ∗ is a cogenerator in R . So, we have the following exact sequence → M g → Q i ∈ I R ∗ , and hence there exists a morphism F h → Q i ∈ I R ∗ such that hf = g and so f is monic. (2) = ⇒ (3) Let E be an injective right R -module. By assumption, let f : E → F be a monic WF nk ( R op ) -preenvelope of E . Therefore, the split exact sequence → E → F → FE → exists,and so E is a direct summand of F . Hence by Proposition 3.3 and [26, Proposition 7.6], E is in WF nk ( R op ) . (3) = ⇒ (1) By assumption, R ∗ is in WF nk ( R op ) , since R ∗ is injective. So, R R is in WI nk ( R ) by Proposition 3.6(2). (3) = ⇒ (4) Let F be a flat left R - module. Then by [27, Theorem 3.52], F ∗ is injective and so F ∗ is in WF nk ( R op ) by assumption, and hence F is in WI nk ( R ) by Proposition 3.6(2). (4) = ⇒ (5) and (5) = ⇒ (1) are clear. (6) = ⇒ (1) By assumption, R R has an epimorphism WI nk ( R ) -cover f : D → R . Then we havea split exact sequence → Ker f → D → R → with D is in WI nk ( R ) . So, by Proposition 3.2and also in particular by [26, Proposition 7.22], R R is in WI nk ( R ) .
1) = ⇒ (6) First, we show that if { M i } i ∈ I is a family of left R -modules and M i is in WI nk ( R ) ,then L i ∈ I M i is in WI nk ( R ) . If k = 0 , then by Proposition 2.10, L i ∈ I M i is n -weak injec-tive if and only if each M i is n -weak injective. The short exact sequence → M i → E i → N i → , where E i is injective exists. Consequently, the sequence → L i ∈ I M i → L i ∈ I E i → L i ∈ I N i → is also exact. If M i is in WI nk ( R ) , then by Proposition 3.2, N i is in WI nk − ( R ) . Byinduction hypothesis, L i ∈ I N i is in WI nk − ( R ) and then by Proposition 3.2 again, L i ∈ I M i is in WI nk ( R ) . On the other hand, by Theorem 4.4, there is a WI nk ( R ) -cover ψ : X → M for any left R -module M . Also, there is an exact sequence → K → P h → M → of left R -modules, where P is an R -module free. Since R R is in WI nk ( R ) , then it follows that P = L i ∈ I R is WI nk − ( R ) .So, there exists a map g : P → X such that ψg = h . Since h is epic, we deduce that ψ : X → M is also epic.Now we define n -super finitely presented dimension of rings. Definition 4.9.
Let R be a ring. Then, l . nsp . gldim( R ) = sup { pd R ( K n − ) | K n − is a special super f initely presented } . It is clear that l . n . sp . gldim( R ) ≤ l . sp . gldim( R ) for any n ≥ . If n = 0 , then l . n . sp . gldim( R ) =l . sp . gldim( R ) . In examples 2.4 and 2.5, since R is coherent, l . sp . gldim( R ) = 2 by [34, Theorem3.8]. But, l . n . sp . gldim( R ) ≤ for any n ≥ , since pd R ( U ) ≤ for every n -super finitelypresented U . Proposition 4.10.
Let R be a ring. Then, the following statements are equivalent: (1) Every right R -module has an epic WF nk ( R op ) -envelope; (2) M is in WI nk +1 ( R ) for every left R -module M ; (3) N is in WF nk +1 for every right R -module N ; (4) Every R -module has a monic WI nk ( R ) -cover; (5) Every quotient of any n -weak injective left R -module is in WI nk ( R ) ; (6) Every submodule of any n -weak flat right R -module is in WF nk ( R op ) ; (7) The kernel of any WI nk ( R ) -precover of any left R -module is in WI nk ( R ) ; The cokernel of any WF nk ( R op ) -preenvelope of any right R -module is in WF nk ( R op ) ; (9) l . nsp . gldim( R ) ≤ k + 1 .Proof. (1) ⇐⇒ (6) Consider the class WF nk ( R op ) of modules with n -weak flat dimensions atmost k . Then, similarly to the proofs of the Proposition 4.1 and Theorem 4.5, the class WF nk ( R op ) is closed under direct summands and direct products, respectively. So [5, Theorem 2] shows that(1) and (6) are equivalent. (4) ⇐⇒ (5) Consider the class WI nk ( R ) of left modules with n -weak injective dimensions atmost k . Then, similar to the proofs of the Propositions 4.2 and 4.8( (1) = ⇒ (6) ), the class WI nk ( R ) is closed under direct summands and direct sums, respectively. Thus from [13, Proposition 4], itfollows that (4) and (5) are equivalent. (6) = ⇒ (5) Let N be a submodule of n -weak injective left R -module M . Then, the short exactsequence → N → M → MN → induces the exactness of → ( MN ) ∗ → M ∗ → N ∗ → . By Proposition 2.6(2), M ∗ is n -weak flat, and hence by hypothesis, ( MN ) ∗ is in WF nk ( R op ) .Consequently, using Proposition 3.6(2), we conclude that MN is in WI nk ( R ) . (5) = ⇒ (6) is similar to the proof of (6) = ⇒ (5) using Propositions 2.6(1 ) and 3.6(1). (1) = ⇒ (8) Let M be a right R -module. Then by Theorem 4.5, there is a WF nk ( R op ) -preenvelope. ψ : M → D . Also by hypothesis, if the map φ : M → Y is an epic WF nk ( R op ) -envelope of M ,then from [9, Lemma 8.6.3], it follows that L ⊕ Y ∼ = D , where L = Coker ψ . So L is in WF nk ( R op ) as a direct summand of D . (8) = ⇒ (6) Consider, the short exact sequence → L → M → D → , where M is n -weakflat. We claim that L is in WF nk ( R op ) . Indeed, we have the following commutative diagram: / / L / / M (cid:15) (cid:15) L h / / X / / Y / / Where h : L → X is a WF nk ( R op ) -preenvelope of L and Y = Coker h . In particular, the sequence → L → X → Y → is exact, and so by Corollary 3.4(2), L is in WF nk ( R op ) . (5) = ⇒ (2) For every left R -module M , there is an exact sequence → M → E → D → ofleft R -modules, where E is injective. By (5), D is in WI nk ( R ) and so by Proposition 3.2, M is in WI nk +1 ( R ) . (2) = ⇒ (5) is clear. ⇐⇒ (3) ⇐⇒ (9) are clear by Proposition 3.5.If k = 0 in WI nk ( R ) and WF nk ( R op ) , then we have the following result: Theorem 4.11.
Let R be a ring. Then, the following statements are equivalent: (1) R R is in WI n ( R ) ; (2) Every left R -module is in WI n ( R ) ; (3) Every special super finitely presented left R -module is in WI n ( R ) ; (4) The short exact sequence → K n → F n → K n − → is a split superpure sequence; (5) Every right R -module is in WF n ( R op ) .Proof. (2) = ⇒ (3) and (2) = ⇒ (1) are trivial. (1) = ⇒ (2) Let N be a left R -module. Consider P → N → where P is free. Since R R is in WI n ( R ) , by Proposition 2.10, we get that P is in WI n ( R ) , and so by Proposition 4.10, N is in WI n ( R ) . (3) = ⇒ (4) Let U be an n -super finitely presented left R -module with special super finitelypresented K n − . Then, we have the short exact sequence → K n → F n → K n − → . Since U is also ( n + 1) -super finitely presented, it follows that K n is special super finitely presented.By assumption, K n is in WI n ( R ) and thus by Remark 2.3(1), Ext R ( K n − , K n ) = 0 and so byProposition 2.8, the above sequence is split superpure. (4) = ⇒ (5) Let the short exact sequence → K n → F n → K n − → be split superpure.Then K n − is flat as a direct summand of F n . Consequently, Tor R ( M, K n − ) = 0 for any right R -module M , and so by Remark 2.3(1), M is in WF n ( R op ) . (5) = ⇒ (2) Let M be any left R -module. Then, M ∗ is a right R -module and hence by as-sumption, M ∗ is in WF n ( R op ) . Therefore by Proposition 2.6(2), every left R -module M is WI n ( R ) .A cotorsion pair (or orthogonal theory of Ext ) consists of a pair ( F , C ) of classes of R -modules[30, 28] such that C = F ⊥ and F = ⊥ C where for a class S , we have S ⊥ = { M : M is an R - module and Ext R ( S, M ) = 0 for all S ∈ S} and ⊥ S = { M : M is an R - module and Ext R ( M, S ) =0 for all S ∈ S} . n [10], Eklof and Trlifaj proved that a cotorsion pair ( F , C ) in R -Mod is complete when itis cogenerated by a set. This result actually holds in any Grothendieck category with enoughprojectives, as Hovey proved in [17].A cotorsion theory ( F , C ) is called hereditary, if whenever → F ′ → F → F ′′ → is exactwith F, F ′′ ∈ F then F ′ is also in F , or equivalently, if → C ′ → C → C ′′ → is anexact sequence with C, C ′ ∈ C , then C ′′ is also in C . A cotorsion pair ( F , C ) is called completeprovided that for any R -module M , there exist exact sequences → M → C → D → and → C ′ → D ′ → M → of R -modules with C, C ′ ∈ C and D, D ′ ∈ F , for more details, see[14, 36]. Proposition 4.12. (1) If n - wid R ( R ) ≤ k , then the pair ( WI nk ( R ) , WI nk ( R ) ⊥ ) is a perfectcotorsion pair. (2) The pair ( ⊥ WI nk ( R ) , WI nk ( R )) is a hereditary cotorsion pair.Proof. (1) The pair ( WI nk ( R ) , WF nk ( R op )) is a duality pair by Proposition 4.1. Similar to theproof of the Proposition 4.1, we show that WI n ( R ) is closed under direct sums and under exten-sions. Also by hypothesis and Proposition 3.3, R is in WI nk ( R ) , and so ( WI nk ( R ) , WF nk ( R op )) isa perfect duality pair. Therefore by Lemma 4.3, it follows that ( WI nk ( R ) , WI nk ( R ) ⊥ ) is a perfectcotorsion pair.(2) First, we show that ( ⊥ WI nk ( R )) ⊥ = WI nk ( R ) . It is clear that WI nk ( R ) ⊆ ( ⊥ WI nk ( R )) ⊥ .Let M is in ( ⊥ WI nk ( R )) ⊥ and U be an n -super finitely presented with special super finitely pre-sented K n − . Then, it follows that K n − is in ⊥ WI nk ( R ) and consequently, Ext R ( K n − , M ) = 0 .Thus by Remark 2.3(1), Ext nR ( U, M ) = 0 and hence M is in WI nk ( R ) .Assume that → M → M → M → is a short exact sequence left R -modules with M , M in WI nk ( R ) , then from Corollary 3.4(1), we conclude that M is in WI nk ( R ) , and so ( ⊥ WI nk ( R ) , WI nk ( R )) is a hereditary cotorsion pair. Proposition 4.13.
The pair ( WF nk ( R op ) , WF nk ( R op ) ⊥ ) is a hereditary perfect cotorsion pair.Proof. The pair ( WF nk ( R op ) , WI nk ( R )) is a duality pair by Proposition 4.2. The class WF nk ( R op ) is closed under direct sums and extensions. By Remark 2.3(5), R is in WF nk ( R op ) , and hence ( WF nk ( R op ) , WI nk ( R )) is a perfect duality pair. Therefore by Lemma 4.3, we deduce that ( WF nk ( R op ) , WF nk ( R op ) ⊥ ) is a perfect cotorsion pair. lso, assume that → M → M → M → is a short exact sequence right R -modules with M , M are in WF nk ( R op ) , then from Corollary 3.4(2), we get that M is in WF nk ( R op ) , and so ( WF nk ( R op ) , WF nk ( R op ) ⊥ ) is a hereditary cotorsion pair.Let ( A , B ) and ( C , D ) are two cotorsion pairs. Then by [36, Remark 4.12], ( A , B ) (cid:22) ( C , D ) if B ⊆ D . By [17, Definition 6.1], the pair ( M , N ) is said to be cogenerated by a set if there is a setof objects M ∈ M such that N ∈ N if and only if Ext R ( M, N ) = 0 for all M ∈ M . Then byRemark 2.3, we have the following easy observations: Remark 4.14. (1)
Let S P res ∞ n be a subclass of all the special n -super finitely presented left R -modules. Then, ( ⊥ WI n ( R ) , WI n ( R )) is a hereditary complete cotorsion pair, since itis cogenerated by a set of representatives for S P res ∞ n . (2) There is a serie of hereditary complete cotorsion pairs for any n ≥ and k ≥ as follows: ( ⊥ WI nk ( R ) , WI nk ( R )) (cid:22) ( ⊥ WI n +1 k ( R ) , WI n +1 k ( R )) (cid:22) ( ⊥ WI n +2 k ( R ) , WI n +2 k ( R )) (cid:22) · · · (3) There is a serie of hereditary cotorsion pairs for any n ≥ and k ≥ as follows: ( WF nk ( R op ) , WF nk ( R op ) ⊥ ) (cid:22) ( WF n +1 k ( R op ) , WF n +1 k ( R op ) ⊥ ) (cid:22) · · · References [1] N. Bourbaki,
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