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
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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 C⊗SN 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 Q∈ModR 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 M∈ModR is defined as inf{n≥0∣Ext≥n+1R(X,M)=0 for any finitely presented left R-module X}, and set FP-idRM=∞ if no such integer exists. For a module B∈ModRop, 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) Ext≥1R(C,C)=0=Ext≥1Sop(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):={P⊗RC∣P is projective in Mod Rop}, |
FC(Sop):={F⊗RC∣F is flat in Mod Rop}, |
IC(S):={I∗∣I is injective in Mod R}, |
FIC(S):={Q∗∣Q is FP−injective 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 N∈FC(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) TorR≥1(N,C)=0.
(a2) Ext≥1Sop(C,N⊗RC)=0.
(a3) The canonical evaluation homomorphism
μN:N→(N⊗RC)∗ |
defined by μN(x)(c)=x⊗c for any x∈N and c∈C 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) Ext≥1R(C,M)=0.
(b2) TorS≥1(C,M∗)=0.
(b3) The canonical evaluation homomorphism
θM:C⊗SM∗→M |
defined by θM(c⊗f)=f(c) for any c∈C and f∈M∗ 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 N∈ModRop (respectively, ModS), N∈AC(Rop) (respectively, AC(S))if and only if N+∈BC(R) (respectively, BC(Sop)).
(2) For a module M∈ModR (respectively, ModSop), M∈BC(R) (respectively, BC(Sop))if and only if M+∈AC(Rop) (respectively, AC(S)).
Let X be a subcategory of ModS. For a module A∈ModS, the X-injective dimension X-idA of A is defined as inf{n≥0∣ there exists an exact sequence
0→A→X0→X1→⋯→Xn→0 |
in ModS with all Xi∈X}, and set X-idA=∞ if no such integer exists. Dually, for a subcategory Y of ModSop and a module B∈ModSop, 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:A⊗SHomR(B,E)→HomR(HomSop(A,B),E) |
in ModT defined by τA,B,E(x⊗f)(g)=fg(x)for any x∈A, f∈HomR(B,E) and g∈HomSop(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+1→⋯→St1→St0→A→0 | (2.1) |
in ModSop with n≥0 and all ti positive integers, then ExtiSop(A,B) is a finitely presented left R-module for any 0≤i≤n.
Proof. Since AS is finitely presented, there exists an exact sequence
St1f0⟶St0→A→0 |
in ModSop with s0,s1 positive integers. Then we get two exact sequences of abelian groups:
St1⊗SHomR(B,E)→St0⊗SHomR(B,E)→A⊗SHomR(B,E)→0, and |
0→HomSop(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 n≥1 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 i≥1. 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 0≤i≤n.
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 N∈ModS with FIC(S)-idN<∞, then N∈AC(S).
Proof. (1) Let E be an FP-injective left R-module. Then Ext≥1R(C,E)=0. By Lemma 3.2, we have
TorSi(C,E∗)≅HomR(ExtiSop(C,C),E)=0 |
for any i≥1. Finally, consider the following sequence of left R-module homomorphisms:
C⊗SE∗τC,C,E⟶HomR(C∗,E)=HomR(R,E)α⟶E, |
where α is the canonical evaluation homomorphism defined by α(h)=h(1R) for any h∈C∗. It is well known that α is an isomorphism with the inverse β:E→HomR(R,E) defined by β(e)(r)=re for any e∈E and r∈R. Note that the unit 1R of R coincides with the identity homomorphism idC of CS. So, for any x∈C and f∈E∗, we have
ατC,C,E(x⊗f)=τC,C,E(x⊗f)(1R)=τC,C,E(x⊗f)(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 E∈BC(R).
(2) Let Q∈FIC(S). Then Q=E∗ for some FP-injective left R-module E. By (1) and [8,Proposition 4.1], we have Q∈AC(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 N∈ModS. Then
FP-idRC⊗SN≤FIC(S)-idN |
with equality when N∈AC(S).
Proof. Let N∈ModS with FIC(S)-idN=n<∞. Then there exists an exact sequence
0→N→E0∗→E1∗→⋯→En∗→0 | (2.3) |
in ModS with all Ei being FP-injective left R-modules. By Proposition 3.3(2), we have Ei∗∈AC(S) and TorS≥1(C,Ei∗)=0 for any 0≤i≤n. Then applying the functor C⊗S− to the exact sequence (2.3) yields the following exact sequence
0→C⊗SN→C⊗SE0∗→C⊗SE1∗→⋯→C⊗SEn∗→0 |
in ModR. By Proposition 3.3(1), we have that Ei∈BC(R) and C⊗SEi∗≅Ei is FP-injective for any 0≤i≤n. Thus FP-idRC⊗SN≤n.
Now suppose N∈AC(S). Then N≅(C⊗SN)∗ and Ext≥1R(C,C⊗SN)=0. If FP-idRC⊗SN=n<∞, then there exists an exact sequence
0→C⊗SN→E0→E1→⋯→En→0 |
in ModR with all Ei being FP-injective. Applying the functor HomR(C,−) to it yields the following exact sequence
0→(C⊗SN)∗(≅N)→E0∗→E1∗→⋯→En∗→0 |
in ModS with all Ei∗∈FIC(S), and so FIC(S)-idN≤n.
We also need the following lemma.
Lemma 3.5.
(1) For any M∈ModRop, we have (M⊗RC)++≅M++⊗RC.
(2) For any N∈ModS, we have (C⊗SN)++≅C⊗SN++.
Proof. (1) By [4,Lemma 2.16(a)(c)], we have (M⊗RC)++≅[(M+)∗]+≅M++⊗RC.
Symmetrically, we get (2).
The following observation is useful in the next section.
Proposition 3.6.
(1) For any N∈ModS, if N+∈FC(Sop), then N∈FIC(S).
(2) For any N∈ModS, if N++∈IC(S), then N∈FIC(S).
(3) For any Q∈ModSop, if Q++∈FC(Sop), then Q∈FC(Sop).
Proof. (1) Let N+∈FC(Sop). Then N+∈BC(Sop) and N∈AC(S) by [8,Corollary 6.1] and Lemma 2.4(1). On the other hand, N++∈IC(S) by Lemma 2.2(1). Then C⊗SN++ is an injective left R-module by [8,Lemma 5.1(c)]. Since (C⊗SN)++≅C⊗SN++ by Lemma 3.5(2), it follows that (C⊗SN)++ is also an injective left R-module. Notice that C⊗SN is a pure submodule of (C⊗SN)++ by [2,Proposition 5.3.9], so C⊗SN is an FP-injective left R-module by [18,Lemma 4]. Since N∈AC(S), we have N≅(C⊗SN)∗∈FIC(S).
(2) It follows from Lemma 2.2(1) and (1).
(3) Let Q∈ModSop. Note that Q is a pure submodule of Q++ by [2,Proposition 5.3.9]. Thus, if Q++∈FC(Sop), then Q∈FC(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 N∈ModS.
(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 N∈ModS. Then for any finitely presented left R-module A and i≥1, we have
TorRi((N+)∗,A)≅TorRi((C⊗SN)+,A)≅[ExtiR(A,C⊗SN)]+ |
by [4,Lemma 2.16(a)(d)], and so TorRi((N+)∗,A)=0 if and only if ExtiR(A,C⊗SN)=0. It implies
fdRop(N+)∗=FP−idRC⊗SN. | (3.1) |
By Proposition 2.3(2) and Lemma 2.4(1), we have that if FIC(S)−idN<∞, then N∈AC(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 N∈AC(S). Then for any n≥0, we have
FIC(S)−idN=n⇔FP−idRC⊗N=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 N∈FIC(S) by Proposition 2.6(1). Conversely, if N∈FIC(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 Q∈FC(Sop) by Proposition 3.6(3). Conversely, if Q∈FC(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)++]I∈FC(Sop) by [8,Proposition 5.1(a)]. By [2,Theorem 3.2.22] and Lemma 3.5(1), we have
[(RR)++]I⊗RC≅[(RR)++⊗RC]I≅[(R⊗RC)++]I≅[(CS)++]I∈FC(Sop). |
Since RR∈AC(Rop), both (RR)++ and [(RR)++]I are in AC(Rop) by Lemma 2.4 and [8,Proposition 4.2(a)]. So [(RR)++]I≅([(RR)++]I⊗RC)∗ 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 U∈FIC(S), there exists a module N∈IC(S) such thatU+ is a direct summand of N+.
Proof. Let U∈FIC(S) such that U=E∗ with E being FP-injective in ModR. Then there exists a pure exact sequence
0→E→I→L→0 |
in ModR with I being injective. By [2,Proposition 5.3.8], the induced exact sequence
0→L+→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 N∈ModS.
(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 U∈FIC(S). By Lemma 4.2, there exists a module N∈IC(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 {Ei∣i∈I} be a family of injective left R-modules with I any index set. By [4,Lemma 2.7], we have
[(⊕i∈IEi)∗]+≅[⊕i∈I(Ei)∗]+≅Πi∈I[(Ei)∗]+. |
Since R is a left coherent ring and since all [(Ei)∗]+ are in FC(Sop) by (3), we have that Πi∈I[(Ei)∗]+, and hence [(⊕i∈IEi)∗]+, is also in FC(Sop) by [8,Proposition 5.1(a)]. Then (⊕i∈IEi)∗∈IC(S) by (3) again. Since all Ei are in BC(R), we have ⊕i∈IEi∈BC(R) by [8,Proposition 4.2(a)]. It follows from [8,Lemma 5.1(c)] that ⊕i∈IEi≅C⊗S(⊕i∈IEi)∗ 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 n≥0. Then the subcategory of ModS consisting of modules Nwith FIC(S)-idN≤n is closed pure submodules and pure quotients.
(2) Let R be a left Noetherian ring and n≥0. Then the subcategory of ModS consisting of modules Nwith IC(S)-idN≤n is closed pure submodules and pure quotients.
Proof. (1) Let
0→K→N→L→0 |
be a pure exact sequence in ModS with FIC(S)-pdN≤n. Then by [2,Proposition 5.3.8], the induced exact sequence
0→L+→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)-pdK≤n and FIC(S)-pdL≤n 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 N∈ModS.
(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 Q∈FC(Sop) by Proposition 2.6(3). Conversely, if Q∈FC(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 N∈ModS.
(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 N∈IC(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, E∗∈IC(S) by (3) again. It follows from Lemma 3.1 and [8,Lemma 5.1(c)] that E≅C⊗SE∗ 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.
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1. | Ya-Nan Li, Zhaoyong Huang, Homological dimensions under Foxby equivalence, 2025, 48, 0386-5991, 10.2996/kmj48102 |