Loading [MathJax]/jax/output/SVG/jax.js
Research article

On the C-flatness and injectivity of character modules

  • Received: 24 October 2021 Revised: 28 March 2022 Accepted: 19 April 2022 Published: 30 May 2022
  • Let R,S be arbitrary associative rings and RCS a semidualizing bimodule. We give some equivalent characterizations for R being left coherent (and right perfect) rings, left Noetherian rings and left Artinian rings in terms of the C-(FP-)injectivity, flatness and projectivity of character modules of certain left S-modules.

    Citation: Zhaoyong Huang. On the C-flatness and injectivity of character modules[J]. Electronic Research Archive, 2022, 30(8): 2899-2910. doi: 10.3934/era.2022147

    Related Papers:

    [1] Yaguo Guo, Shilin Yang . Projective class rings of the category of Yetter-Drinfeld modules over the $ 2 $-rank Taft algebra. Electronic Research Archive, 2023, 31(8): 5006-5024. doi: 10.3934/era.2023256
    [2] Yajun Ma, Haiyu Liu, Yuxian Geng . A new method to construct model structures from left Frobenius pairs in extriangulated categories. Electronic Research Archive, 2022, 30(8): 2774-2787. doi: 10.3934/era.2022142
    [3] Hongjian Li, Kaili Yang, Pingzhi Yuan . The asymptotic behavior of the reciprocal sum of generalized Fibonacci numbers. Electronic Research Archive, 2025, 33(1): 409-432. doi: 10.3934/era.2025020
    [4] Dongxing Fu, Xiaowei Xu, Zhibing Zhao . Generalized tilting modules and Frobenius extensions. Electronic Research Archive, 2022, 30(9): 3337-3350. doi: 10.3934/era.2022169
    [5] Zhefeng Xu, Xiaoying Liu, Luyao Chen . Hybrid mean value involving some two-term exponential sums and fourth Gauss sums. Electronic Research Archive, 2025, 33(3): 1510-1522. doi: 10.3934/era.2025071
    [6] Kuo-Chih Hung, Shin-Hwa Wang, Jhih-Jyun Zeng . Ordering properties of positive solutions for a class of $ \varphi $-Laplacian quasilinear Dirichlet problems. Electronic Research Archive, 2022, 30(5): 1918-1935. doi: 10.3934/era.2022097
    [7] Licui Zheng, Dongmei Li, Jinwang Liu . Some improvements for the algorithm of Gröbner bases over dual valuation domain. Electronic Research Archive, 2023, 31(7): 3999-4010. doi: 10.3934/era.2023203
    [8] Die Hu, Peng Jin, Xianhua Tang . The existence results for a class of generalized quasilinear Schrödinger equation with nonlocal term. Electronic Research Archive, 2022, 30(5): 1973-1998. doi: 10.3934/era.2022100
    [9] Dong Su, Shilin Yang . Representation rings of extensions of Hopf algebra of Kac-Paljutkin type. Electronic Research Archive, 2024, 32(9): 5201-5230. doi: 10.3934/era.2024240
    [10] Fedor Petrov, Zhi-Wei Sun . Proof of some conjectures involving quadratic residues. Electronic Research Archive, 2020, 28(2): 589-597. doi: 10.3934/era.2020031
  • Let R,S be arbitrary associative rings and RCS a semidualizing bimodule. We give some equivalent characterizations for R being left coherent (and right perfect) rings, left Noetherian rings and left Artinian rings in terms of the C-(FP-)injectivity, flatness and projectivity of character modules of certain left S-modules.



    Let R be an arbitrary associative ring and M a left R-module. The right R-module M+:=HomZ(M,Q/Z) is called the character module of M, where Z is the additive group of integers and Q is the additive group of rational numbers. Character modules are a kind of dual modules having nice properties, which played an important role in studying the classification and structure of rings in terms of their modules; see [1,2,3,4,5] and references therein. In particular, Cheatham and Stone [1] gave some equivalent characterizations for a ring R being left coherent (and right perfect), left Noetherian and left Artinian in terms of the (FP-)injectivity, flatness and projectivity of character modules of certain left R-modules.

    On the other hand, the study of semidualizing modules in commutative rings was initiated by Foxby [6] and Golod [7]. Then Holm and White [8] extended it to arbitrary associative rings. Many authors have studied the properties of semidualizing modules and related modules; see [6,7,8,9,10,11,12,13,14,15,16,17], and so on. Among various research areas on semidualizing modules, one basic theme is to extend the "absolute" classical results in homological algebra to the "relative" setting with respect to semidualizing modules. The aim of this paper is to study whether those results of Cheatham and Stone [1] mentioned above have relative counterparts with respect to semidualizing modules. The paper is organized as follows.

    In Section 2, we give some terminology and some preliminary results.

    Let R and S be arbitrary associative rings and RCS a semidualizing bimodule. Assume that R is a left coherent ring. In Section 3, we show that any FP-injective left R-module is in the Bass class BC(R), and any left S-module with finite C-FP-injective dimension is in the Auslander class AC(S) (Proposition 3.3). Then we get that for any module N in ModS, the FP-injective dimension of CSN is at most the C-FP-injective dimension of N, and with equality when N is in AC(S) (Theorem 3.4).

    In Section 4, we show that R is a left coherent (and right perfect) ring if and only if for any left S-module N, the C-FP-injective dimension of N and the C-flat (respectively, C-projective) dimension of N+ are identical, and if and only if (C(I)S)++ is C-flat (respectively, C-projective) for any index set I (Theorems 4.1 and 4.5). Moreover, we get that R is a left Noetherian (respectively, Artinian) ring if and only if for any left S-module N, the C-injectivity of N coincides with the C-flatness (respectively, C-projectivity) of N+ (Theorems 4.3 and 4.6).

    Throughout this paper, all rings are associative rings with unit and all modules are unital. For a ring R, we use ModR to denote the category of left R-modules.

    Recall from [18,19] that a module QModR is called FP-injective (or absolutely pure) if Ext1R(X,Q)=0 for any finitely presented left R-module X. The FP-injective dimension FP-idRM of a module MModR is defined as inf{n0Extn+1R(X,M)=0 for any finitely presented left R-module X}, and set FP-idRM= if no such integer exists. For a module BModRop, we use fdRopB to denote the flat dimension of B.

    Definition 2.1. (see [8,20]). Let R and S be arbitrary rings. An (R-S)-bimodule RCS is called semidualizing if the following conditions are satisfied.

    (1) RC admits a degreewise finite R-projective resolution and CS admits a degreewise finite Sop-projective resolution.

    (2) R=End(CS) and S=End(RC).

    (3) Ext1R(C,C)=0=Ext1Sop(C,C).

    Wakamatsu [15] introduced and studied the so-called generalized tilting modules, which are usually called Wakamatsu tilting modules, see [21,22]. Note that a bimodule RCS is semidualizing if and only if it is Wakamatsu tilting ([17,Corollary 3.2]). Typical examples of semidualizing bimodules include the free module of rank one and the dualizing module over a Cohen-Macaulay local ring. More examples of semidualizing bimodules can be found in [8,13,16].

    From now on, R and S are arbitrary rings and we fix a semidualizing bimodule RCS. We write ():=Hom(C,), and write

    PC(Sop):={PRCP  is projective in Mod  Rop},
    FC(Sop):={FRCF  is flat in Mod  Rop},
    IC(S):={II  is injective in Mod  R},
    FIC(S):={QQ  is FPinjective in Mod  R}.

    The modules in PC(Sop), FC(Sop), IC(S) and FIC(S) are called C-projective, C-flat, C-injective and C-FP-injective respectively. By [10,Proposition 2.4(1)], we have PC(Sop)=AddCS, where AddCS is the subcategory of ModSop consisting of direct summands of direct sums of copies of CS. When RCS=RRR, C-projective, C-flat, C-injective and C-FP-injective modules are exactly projective, flat, injective and FP-injective modules respectively.

    Lemma 2.2. (see [27,Theorem 4.17(1) and Corollary 4.18(1)])

    (1) A right S-module NFC(Sop) if and only if N+IC(S).

    (2) The class FC(Sop) is closed under pure submodules and pure quotients.

    The following definition is cited from [8].

    Definition 2.3.

    (1) The Auslander class AC(Rop) with respect to C consists of all modules N in ModRop satisfying the following conditions.

    (a1) TorR1(N,C)=0.

    (a2) Ext1Sop(C,NRC)=0.

    (a3) The canonical evaluation homomorphism

    μN:N(NRC)

    defined by μN(x)(c)=xc for any xN and cC is an isomorphism in ModRop.

    (2) The Bass class BC(R) with respect to C consists of all modules M in ModR satisfying the following conditions.

    (b1) Ext1R(C,M)=0.

    (b2) TorS1(C,M)=0.

    (b3) The canonical evaluation homomorphism

    θM:CSMM

    defined by θM(cf)=f(c) for any cC and fM is an isomorphism in ModR.

    (3) The Auslander class AC(S) in ModS and the Bass class BC(Sop) in ModSop are defined symmetrically.

    The following lemma will be used frequently in the sequel.

    Lemma 2.4. (see [5,Proposition 3.2])

    (1) For a module NModRop (respectively, ModS), NAC(Rop) (respectively, AC(S))if and only if N+BC(R) (respectively, BC(Sop)).

    (2) For a module MModR (respectively, ModSop), MBC(R) (respectively, BC(Sop))if and only if M+AC(Rop) (respectively, AC(S)).

    Let X be a subcategory of ModS. For a module AModS, the X-injective dimension X-idA of A is defined as inf{n0 there exists an exact sequence

    0AX0X1Xn0

    in ModS with all XiX}, and set X-idA= if no such integer exists. Dually, for a subcategory Y of ModSop and a module BModSop, the Y-projective dimension Y-pdB of B is defined.

    Recall that a ring R is called left coherent if any finitely generated left ideal of R is finitely presented. We begin with the following lemma.

    Lemma 3.1. Let R,S,T be arbitrary rings and consider the situation (TAS,RBS) with AS and RB finitely presented.

    (1) If HomSop(A,B) is a finitely generated left R-module, then for any FP-injective left R-module E, there exists a natural isomorphism

    τA,B,E:ASHomR(B,E)HomR(HomSop(A,B),E)

    in ModT defined by τA,B,E(xf)(g)=fg(x)for any xA, fHomR(B,E) and gHomSop(A,B).

    (2) If R is a left coherent ring, then HomSop(A,B) is a finitely presented left R-module.Moreover, if there exists an exact sequence

    Stn+1St1St0A0 (2.1)

    in ModSop with n0 and all ti positive integers, then ExtiSop(A,B) is a finitely presented left R-module for any 0in.

    Proof. Since AS is finitely presented, there exists an exact sequence

    St1f0St0A0

    in ModSop with s0,s1 positive integers. Then we get two exact sequences of abelian groups:

    St1SHomR(B,E)St0SHomR(B,E)ASHomR(B,E)0, and
    0HomSop(A,B)HomSop(St0,B)HomSop(f0,B)HomSop(St1,B) (2.2)

    with HomSop(Sti,B)Bti being a finitely presented left R-module for i=0,1.

    (1) If HomSop(A,B) is a finitely generated left R-module, then Im(HomSop(f0,B)) and Coker(HomSop(f0,B)) are finitely presented left R-modules by [24,Proposition 1.6(ii)]. Thus for any FP-injective left R-module E, applying the functor HomR(,E) to (2.2) yields the following exact sequence of abelian groups:

    HomR(HomSop(St1,B),E)HomR(HomSop(St0,B),E)HomR(HomSop(A,B),E)0.

    By [28,Lemma 3.55(i)], there exists the following commutative diagram

    with both τSt0,B,E and τSt1,B,E being isomorphisms of abelian groups. So τA,B,E is also an isomorphism of abelian groups. Notice that A is a (T,S)-bimodule, so τA,B,E is an isomorphism of left T-modules.

    (2) If R is a left coherent ring, then Im(HomSop(f0,B)) is a finitely generated left R-submodule of the finitely presented left R-module HomSop(St1,B), and so Im(HomSop(f0,B)) is a finitely presented left R-module. It follows from [24,Proposition 1.6(i)] that HomSop(A,B) is a finitely generated left R-submodule of the finitely presented left R-module HomSop(St0,B), and hence HomSop(A,B) is a finitely presented left R-module.

    Assume that there exists an exact sequence as in (2.1). We prove the latter assertion by induction on n. The case for n=0 follows from the former assertion. Suppose n1 and set K1:=Imf0. Then we get an exact sequence

    HomSop(St0,B)HomSop(K1,B)Ext1Sop(A,B)0

    and an isomorphism

    Exti+1Sop(A,B)ExtiSop(K1,B)

    in ModR for any i1. Now the assertion follows easily from the induction hypothesis.

    The following result is a generalization of [28,Theorem 10.66].

    Lemma 3.2. Let R,S,T be arbitrary rings and consider the situation (TAS,RBS) such thatRB is finitely presented and there exists an exact sequence as in (2.1). If R is a left coherent ring, then for any FP-injectiveleft R-module E, there exists a natural isomorphism

    TorSi(A,HomR(B,E))HomR(ExtiSop(A,B),E)

    in ModT for any 0in.

    Proof. Applying Lemma 3.1, we get the assertion by using an argument similar to that in the proof of [28,Theorem 10.66].

    It was shown in [8,Lemma 4.1] that any injective left R-module is in BC(R). The assertion (1) in the following proposition extends this result.

    Proposition 3.3. Let R be a left coherent ring. Then we have

    (1) Any FP-injective left R-module is in BC(R).

    (2) If NModS with FIC(S)-idN<, then NAC(S).

    Proof. (1) Let E be an FP-injective left R-module. Then Ext1R(C,E)=0. By Lemma 3.2, we have

    TorSi(C,E)HomR(ExtiSop(C,C),E)=0

    for any i1. Finally, consider the following sequence of left R-module homomorphisms:

    CSEτC,C,EHomR(C,E)=HomR(R,E)αE,

    where α is the canonical evaluation homomorphism defined by α(h)=h(1R) for any hC. It is well known that α is an isomorphism with the inverse β:EHomR(R,E) defined by β(e)(r)=re for any eE and rR. Note that the unit 1R of R coincides with the identity homomorphism idC of CS. So, for any xC and fE, we have

    ατC,C,E(xf)=τC,C,E(xf)(1R)=τC,C,E(xf)(idC)=fidC(x)=f(x),

    which implies θE=ατC,C,E. Since τC,C,E is an isomorphism by Lemma 3.1(1), it follows that θE is also an isomorphism. Thus we conclude that EBC(R).

    (2) Let QFIC(S). Then Q=E for some FP-injective left R-module E. By (1) and [8,Proposition 4.1], we have QAC(S). Now the assertion follows from [8,Theorem 6.2].

    Now we are in a position to prove the following result.

    Theorem 2.4. Let R be a left coherent ring and NModS. Then

    FP-idRCSNFIC(S)-idN

    with equality when NAC(S).

    Proof. Let NModS with FIC(S)-idN=n<. Then there exists an exact sequence

    0NE0E1En0 (2.3)

    in ModS with all Ei being FP-injective left R-modules. By Proposition 3.3(2), we have EiAC(S) and TorS1(C,Ei)=0 for any 0in. Then applying the functor CS to the exact sequence (2.3) yields the following exact sequence

    0CSNCSE0CSE1CSEn0

    in ModR. By Proposition 3.3(1), we have that EiBC(R) and CSEiEi is FP-injective for any 0in. Thus FP-idRCSNn.

    Now suppose NAC(S). Then N(CSN) and Ext1R(C,CSN)=0. If FP-idRCSN=n<, then there exists an exact sequence

    0CSNE0E1En0

    in ModR with all Ei being FP-injective. Applying the functor HomR(C,) to it yields the following exact sequence

    0(CSN)(N)E0E1En0

    in ModS with all EiFIC(S), and so FIC(S)-idNn.

    We also need the following lemma.

    Lemma 3.5.

    (1) For any MModRop, we have (MRC)++M++RC.

    (2) For any NModS, we have (CSN)++CSN++.

    Proof. (1) By [4,Lemma 2.16(a)(c)], we have (MRC)++[(M+)]+M++RC.

    Symmetrically, we get (2).

    The following observation is useful in the next section.

    Proposition 3.6.

    (1) For any NModS, if N+FC(Sop), then NFIC(S).

    (2) For any NModS, if N++IC(S), then NFIC(S).

    (3) For any QModSop, if Q++FC(Sop), then QFC(Sop).

    Proof. (1) Let N+FC(Sop). Then N+BC(Sop) and NAC(S) by [8,Corollary 6.1] and Lemma 2.4(1). On the other hand, N++IC(S) by Lemma 2.2(1). Then CSN++ is an injective left R-module by [8,Lemma 5.1(c)]. Since (CSN)++CSN++ by Lemma 3.5(2), it follows that (CSN)++ is also an injective left R-module. Notice that CSN is a pure submodule of (CSN)++ by [2,Proposition 5.3.9], so CSN is an FP-injective left R-module by [18,Lemma 4]. Since NAC(S), we have N(CSN)FIC(S).

    (2) It follows from Lemma 2.2(1) and (1).

    (3) Let QModSop. Note that Q is a pure submodule of Q++ by [2,Proposition 5.3.9]. Thus, if Q++FC(Sop), then QFC(Sop) by Lemma 2.2(2).

    In the following result, we give some equivalent characterizations for R being left coherent in terms of the C-FP-injectivity and flatness of character modules of certain left S-modules, in which the equivalence between (1) and (3) has been obtained in [12,Lemma 4.1] when RCS is faithful.

    Theorem 4.1. The following statements are equivalent.

    (1) R is a left coherent ring.

    (2) FIC(S)-idN=FC(Sop)-pdN+ for any NModS.

    (3) A left S-module N is C-FP-injective (if and) only if N+ is a C-flat right S-module.

    (4) A left S-module N is C-FP-injective (if and) only if N++ is a C-injective left S-module.

    (5) A right S-module Q is C-flat (if and) only if Q++ is a C-flat right S-module.

    (6) If Q is a C-projective right S-module, then Q++ is a C-flat right S-module.

    (7) (C(I)S)++ is a C-flat right S-module for any index set I.

    Proof. (1)(2) Let NModS. Then for any finitely presented left R-module A and i1, we have

    TorRi((N+),A)TorRi((CSN)+,A)[ExtiR(A,CSN)]+

    by [4,Lemma 2.16(a)(d)], and so TorRi((N+),A)=0 if and only if ExtiR(A,CSN)=0. It implies

    fdRop(N+)=FPidRCSN. (3.1)

    By Proposition 2.3(2) and Lemma 2.4(1), we have that if FIC(S)idN<, then NAC(S) and N+BC(Sop). On the other hand, by [8,Corollary 6.1] and Lemma 2.4(1), we have that if FC(Sop)-pdN+<, then N+BC(Sop) and NAC(S). Then for any n0, we have

         FIC(S)idN=nFPidRCN=n   (by Theorem 3.4)fdRop(N+)=n   (by (3.1))FC(Sop)pdN+=n.   (by [14, Lemma 2.6(1)])

    The implications (2)(3) and (5)(6)(7) are trivial.

    (3)(4) If N++IC(S), then N+FC(Sop) by Lemma 2.2(1), and hence NFIC(S) by Proposition 2.6(1). Conversely, if NFIC(S), then N+FC(Sop) by (3), and hence N++IC(S) by Lemma 2.2(1) again.

    (4)(5) If Q++FC(Sop), then QFC(Sop) by Proposition 3.6(3). Conversely, if QFC(Sop), then Q+IC(S) by Lemma 2.2(1). Thus Q+++IC(S) by (4), and therefore Q++FC(Sop) by Lemma 2.2(1) again.

    (7)(1) By [26,Theorem 2.1], it suffices to prove that (RR)I is a flat right R-module for any index set I. By (7), we have [((CS)+)I]+(C(I)S)++FC(Sop). Since there exists a pure monomorphism λ:[(CS)+](I)[(CS)+]I by [1,Lemma 1(1)], it follows from [2,Proposition 5.3.8] that λ+ is a split epimorphism and [(CS)++]I([((CS)+)(I)]+) is a direct summand of [((CS)+)I]+. Then [(CS)++]IFC(Sop) by [8,Proposition 5.1(a)]. By [2,Theorem 3.2.22] and Lemma 3.5(1), we have

    [(RR)++]IRC[(RR)++RC]I[(RRC)++]I[(CS)++]IFC(Sop).

    Since RRAC(Rop), both (RR)++ and [(RR)++]I are in AC(Rop) by Lemma 2.4 and [8,Proposition 4.2(a)]. So [(RR)++]I([(RR)++]IRC) is a flat right R-module by [14,Lemma 2.6(1)]. Since RR is a pure submodule of (RR)++ by [2,Proposition 5.3.9], it follows from [1,Lemma 1(2)] that (RR)I is a pure submodule of [(RR)++]I, and hence (RR)I is also a flat right R-module.

    We need the following lemma.

    Lemma 4.2. For any UFIC(S), there exists a module NIC(S) such thatU+ is a direct summand of N+.

    Proof. Let UFIC(S) such that U=E with E being FP-injective in ModR. Then there exists a pure exact sequence

    0EIL0

    in ModR with I being injective. By [2,Proposition 5.3.8], the induced exact sequence

    0L+I+E+0

    in ModRop splits and E+ is a direct summand of I+. Then E+RC is a direct summand of I+RC. By [4,Lemma 2.16(c)], we have

    U+=(E)+E+RC  and  (I)+I+RC.

    Thus U+(E+RC) is a direct summand of (I)+(I+RC).

    We give some equivalent characterizations for R being left Noetherian in terms of the C-injectivity and flatness of character modules of certain left S-modules as follows.

    Theorem 4.3. The following statements are equivalent.

    (1) R is a left Noetherian ring.

    (2) IC(S)-idN=FC(Sop)-pdN+ for any NModS.

    (3) A left S-module N is C-injective if and only if N+ is a C-flat right S-module.

    (4) A left S-module N is C-injective if and only if N++ is a C-injective left S-module.

    Proof. (1)(2) Let R be a left Noetherian ring. Then a left R-module is FP-injective if and only if it is injective, and so a left S-module is C-FP-injective if and only if it is C-injective. Thus the assertion follows from Theorem 4.1.

    (2)(3) It is trivial.

    By Lemma 2.2(1), we have that for a left S-module N, N+FC(Sop) if and only if N++IC(S). Thus the assertion (3)(4) follows.

    (3)(1) Let UFIC(S). By Lemma 4.2, there exists a module NIC(S) such that U+ is a direct summand of N+. Then U+FC(Sop) by (3) and [8,Proposition 5.1(a)]. Thus R is a left coherent ring by Theorem 4.1.

    To prove that R is a left Noetherian ring, it suffices to prove that the class of injective left R-modules is closed under direct sums by [25,Theorem 2.1]. Let {EiiI} be a family of injective left R-modules with I any index set. By [4,Lemma 2.7], we have

    [(iIEi)]+[iI(Ei)]+ΠiI[(Ei)]+.

    Since R is a left coherent ring and since all [(Ei)]+ are in FC(Sop) by (3), we have that ΠiI[(Ei)]+, and hence [(iIEi)]+, is also in FC(Sop) by [8,Proposition 5.1(a)]. Then (iIEi)IC(S) by (3) again. Since all Ei are in BC(R), we have iIEiBC(R) by [8,Proposition 4.2(a)]. It follows from [8,Lemma 5.1(c)] that iIEiCS(iIEi) is an injective left R-module.

    As a consequence of Theorems 4.1 and 4.3, we get the following corollary, which generalizes [8,Lemma 5.2(c)].

    Corollary 4.4.

    (1) Let R be a left coherent ring and n0. Then the subcategory of ModS consisting of modules Nwith FIC(S)-idNn is closed pure submodules and pure quotients.

    (2) Let R be a left Noetherian ring and n0. Then the subcategory of ModS consisting of modules Nwith IC(S)-idNn is closed pure submodules and pure quotients.

    Proof. (1) Let

    0KNL0

    be a pure exact sequence in ModS with FIC(S)-pdNn. Then by [2,Proposition 5.3.8], the induced exact sequence

    0L+N+K+0

    in ModSop splits and both K+ and L+ are direct summands of N+. By Theorem 3.1, we have FC(Sop)-pdN+n. Note that the class of right S-modules with FC(Sop)-projective dimension at most n is closed under direct summands by [27,Corollary 4.18(1)]. It follows that FC(Sop)-pdK+n and FC(Sop)-pdL+n. Thus FIC(S)-pdKn and FIC(S)-pdLn by Theorem 3.1 again.

    (2) From the proof of (1)(2) in Theorem 4.3, we know that if R is a left Noetherian ring, then FIC(S)=IC(S). Now the assertion follows from (1).

    In the following result, we give some equivalent characterizations for R being left coherent and right perfect in terms of the C-FP-injectivity and projectivity of character modules of certain left S-modules.

    Theorem 4.5. The following statements are equivalent.

    (1) R is a left coherent and right perfect ring.

    (2) FIC(S)-idN=PC(Sop)-pdN+ for any NModS.

    (3) A left S-module N is C-FP-injective (if and) only if N+ is a C-projective right S-module.

    (4) A right S-module Q is C-flat (if and) only if Q++ is a C-projective right S-module.

    (5) If Q is a C-projective right S-module, then Q++ is a C-projective right S-module.

    (6) (C(I)S)++ is a C-projective right S-module for any index set I.

    Proof. (1)(2) Let R be a left coherent and right perfect ring. Then a right R-module is flat and only if it is projective by [23,Theorem 28.4], and hence FC(Sop)=PC(Sop). Thus the assertion follows from Theorem 3.1.

    The implications (2)(3) and (4)(5)(6) are trivial.

    (3)(4) If Q++PC(Sop), then QFC(Sop) by Proposition 2.6(3). Conversely, if QFC(Sop), then Q+IC(S) by Lemma 2.2(1), and hence Q++PC(Sop) by (3).

    (6)(1) It follows from (6) and Theorem 3.1 that R is a left coherent ring. Let I be an infinite set such that its cardinality is greater than the cardinality of R. By using an argument similar to that in the proof (7)(1) in Theorem 3.1, we get that [(RR)++]I is a projective right R-module and (RR)I is a pure submodule of [(RR)++]I, and hence (RR)I is a pure submodule of a free right R-module. It follows from [26,Theorems 3.1 and 3.2] that R is a right perfect ring.

    Observe from [23,Corollary 15.23 and Theorem 28.4] that R is a left Artinian ring if and only if R is a left Noetherian and right (or left) perfect ring. Finally, we give some equivalent characterizations for R being left Artinian in terms of the C-injectivity and projectivity of character modules of certain left S-modules as follows.

    Theorem 4.6. The following statements are equivalent.

    (1) R is a left Artinian ring.

    (2) IC(S)-idN=PC(Sop)-pdN+ for any NModS.

    (3) A left S-module N is C-injective if and only if N+ is a C-projective right S-module.

    Proof. The implication (2)(3) is trivial.

    If R is a left Artinian ring, then FIC(S)=IC(S) and FC(Sop)=PC(Sop). Thus the implication (1)(2) follows from Theorems 4.3 and 4.5.

    (3)(1) Let E be an FP-injective left R-module. Then by Lemma 3.2, there exists a module NIC(S) such that (E)+ is a direct summand of N+. So (E)+PC(Sop) by (3) and [8,Proposition 5.1(b)], and hence R is a left coherent and right perfect ring by Theorem 4.5.

    On the other hand, EIC(S) by (3) again. It follows from Lemma 3.1 and [8,Lemma 5.1(c)] that ECSE is an injective left R-module. Then R is left Noetherian ring by [18,Theorem 3]. Thus we conclude that R is a left Noetherian and right perfect ring, and hence a left Artinian ring.

    The research was partially supported by NSFC (Grant Nos. 11971225, 12171207).

    The author declares there is no conflicts of interest.



    [1] T. J. Cheatham, D. R. Stone, Flat and projective character modules, Proc. Amer. Math. Soc., 81 (1981), 175–177. https://doi.org/10.1090/S0002-9939-1981-0593450-2 doi: 10.1090/S0002-9939-1981-0593450-2
    [2] E. E. Enochs, O. M. G. Jenda, Relative Homological Algebra, Vol. 1, the second revised and extended edition, de Gruyter Expositions in Math, Walter de Gruyter GmbH & Co. KG, Berlin, 2011. https://doi.org/10.1515/9783110215212
    [3] D. J. Fieldhouse, Character modules, Comment. Math. Helv., 46 (1971), 274–276. https://doi.org/10.1007/BF02566844
    [4] R. Göbel, J. Trlifaj, Approximations and Endomorphism Algebras of Modules, Second revised and extended edition, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. https://doi.org/10.1515/9783110218114
    [5] Z. Y. Huang, Duality pairs induced by Auslander and Bass classes, Georgian Math. J., 28 (2021), 867–882. https://doi.org/10.1515/gmj-2021-2101 doi: 10.1515/gmj-2021-2101
    [6] H. B. Foxby, Gorenstein modules and related modules, Math. Scand., 31 (1972), 267–284. https://doi.org/10.7146/math.scand.a-11434 doi: 10.7146/math.scand.a-11434
    [7] E. S. Golod, G-dimension and generalized perfect ideals, Trudy Mat. Inst. Steklov., 165 (1984), 62–66.
    [8] H. Holm, D. White, Foxby equivalence over associative rings, J. Math. Kyoto Univ., 47 (2007), 781–808. https://doi.org/10.1215/kjm/1250692289 doi: 10.1215/kjm/1250692289
    [9] H. Holm, P. Jøgensen, Semidualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra, 205 (2006), 423–445. https://doi.org/10.1016/j.jpaa.2005.07.010 doi: 10.1016/j.jpaa.2005.07.010
    [10] Z. F. Liu, Z. Y. Huang, A. M. Xu, Gorenstein projective dimension relative to a semidualizing bimodule, Comm. Algebra, 41 (2013), 1–18. https://doi.org/10.1080/00927872.2011.602782 doi: 10.1080/00927872.2011.602782
    [11] R. Takahashi, D. White, Homological aspects of semidualizing modules, Math. Scand., 106 (2010), 5–22. https://doi.org/10.7146/math.scand.a-15121 doi: 10.7146/math.scand.a-15121
    [12] X. Tang, FP-injectivity relative to a semidualizing bimodule, Publ. Math. Debrecen, 80 (2012), 311–326. https://doi.org/10.5486/PMD.2012.4907 doi: 10.5486/PMD.2012.4907
    [13] X. Tang, Z. Y. Huang, Homological aspects of the dual Auslander transpose II, Kyoto J. Math., 57 (2017), 17–53. https://doi.org/10.1215/21562261-3759504 doi: 10.1215/21562261-3759504
    [14] X. Tang, Z. Y. Huang, Homological invariants related to semidualizing bimodules, Colloq. Math., 156 (2019), 135–151. https://doi.org/10.4064/cm7476-3-2018 doi: 10.4064/cm7476-3-2018
    [15] T. Wakamatsu, On modules with trivial self-extensions, J. Algebra, 114 (1988), 106–114. https://doi.org/10.1016/0021-8693(88)90215-3 doi: 10.1016/0021-8693(88)90215-3
    [16] T. Wakamatsu, Stable equivalence for self-injective algebras and a generalization of tilting modules, J. Algebra, 134 (1990), 298–325. https://doi.org/10.1016/0021-8693(90)90055-S doi: 10.1016/0021-8693(90)90055-S
    [17] T. Wakamatsu, Tilting modules and Auslander's Gorenstein property, J. Algebra, 275 (2004), 3–39. https://doi.org/10.1016/j.jalgebra.2003.12.008 doi: 10.1016/j.jalgebra.2003.12.008
    [18] B. H. Maddox, Absolutely pure modules, Proc. Amer. Math. Soc., 18 (1967), 155–158. https://doi.org/10.1090/S0002-9939-1967-0224649-5
    [19] B. Stenström, Coherent rings and FP-injective modules, J. London Math. Soc., 2 (1970), 323–329. https://doi.org/10.1112/jlms/s2-2.2.323 doi: 10.1112/jlms/s2-2.2.323
    [20] T. Araya, R. Takahashi, Y. Yoshino, Homological invariants associated to semi-dualizing bimodules, J. Math. Kyoto Univ., 45 (2005), 287–306. https://doi.org/10.1215/kjm/1250281991 doi: 10.1215/kjm/1250281991
    [21] A. Beligiannis, I. Reiten, Homological and homotopical aspects of Torsion theories, Amer. Math. Soc., 188 (2007). https://doi.org/http://dx.doi.org/10.1090/memo/0883
    [22] F. Mantese, I. Reiten, Wakamatsu tilting modules, J. Algebra, 278 (2004), 532–552. https://doi.org/10.1016/j.jalgebra.2004.03.023
    [23] F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, Secondnd edition, Springer-Verlag, New York-Berlin-Heidelberg, 1992. https://doi.org/10.1007/978-1-4612-4418-9
    [24] G. Azumaya, Countable generatedness version of rings of pure global dimension zero, in Representations of Algebras and Related Topics, London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge, (1992), 43–79. https://doi.org/10.1017/CBO9780511661853.003
    [25] H. Bass, Injective dimension in Noetherian rings, Trans. Amer. Math. Soc., 102 (1962), 18–29. https://doi.org/10.2307/1993878 doi: 10.2307/1993878
    [26] S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc., 97 (1960), 457–473. https://doi.org/10.2307/1993382
    [27] Z. Y. Huang, Homological dimensions relative to preresolving subcategories II, Forum Math., 34 (2022), 507–530. https://doi.org/10.1515/forum-2021-0136 doi: 10.1515/forum-2021-0136
    [28] J. J. Rotman, An Introduction to Homological Algebra, Second edition, Universitext, Springer, New York, 2009. https://doi.org/10.1007/b98977
  • This article has been cited by:

    1. Ya-Nan Li, Zhaoyong Huang, Homological dimensions under Foxby equivalence, 2025, 48, 0386-5991, 10.2996/kmj48102
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1725) PDF downloads(65) Cited by(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog