FP-INJECTIVE AND WEAKLY QUASI-FROBENIUS RINGS

GRIGORY GARKUSHA

Abstract. The classes of F P -injective and weakly quasi-Frobenius rings are investigated. The properties for both classes of rings are closely linked with embedding of finitely presented modules in fp-flat and free modules respectively. Using these properties, we characterize the classes of coherent CF and FGF-rings. Moreover, it is proved that the group ring R(G) is F P -injective (weakly quasi-Frobenius) if and only if the ring R is F P -injective (weakly quasi-Frobenius) and the group G is locally finite.

0. Introduction

An application of the duality context with respect to the bimodule _{R}R_{R} to the categories of finitely generated left and right R-modules leads to the case when R is a noetherian self-injective ring. Such rings are called quasi-Frobenius (or QF-rings). In turn, an R-duality for categories of finitely presented modules leads to the class of weakly quasi-Frobenius rings (or WQF-rings). Such rings can be described as coherent F P -injective rings [1].

In the present paper we continue an investigation of the classes of F P - injective and WQF-rings. To begin with, one must introduce a notion of an F P -cogenerator, which plays an essential role in our analysis, aproximately the same one as the notion of the cogenerator for the class of QF-rings. Using also properties of fp-flat and fp-injective modules, we give new criteria for both classes of rings (theorems 2.2, 2.8, and 2.9), which allow to describe also the classes of coherent CF and FGF-rings. Moreover, it is proved analogs of Renault’s and Connell’s theorems for the F P -injective and weakly quasi-Frobenius group rings respectively (theorems 3.2 and 3.5).

It should be emphasized that the most difficult with the technical point of view statements for the F P -injective rings are proved with the help of the category of generalized R-modules _{R}C = (mod −R, Ab) which consist of additive covariant functors from the category of finitely presented right Rmodules mod −R to the category of abelian groups Ab. In our situation this is the typical case since it is localizing subcategories of the category _{R}C and corresponding to them torsion functors enable to adapt many properties we are interested in of the category of modules to the category of finitely presented modules. It is with the latter category the most interesting statements for F P -injective and WQF-rings are linked.

Throughout the paper the category of left (respectively right) R-modules is denoted by R − Mod (respectively Mod −R) and the category of finitely presented left (respectively right) R-modules by R−mod (respectively mod −R).

2 Grigory Garkusha

The dual module Hom_{R}(M, R) of M ∈ R − Mod is denoted by M^{∗}. Regular rings are supposed to be von Neumann regular.

I should like to thank A. I. Generalov for some helpful discussions.

1. Preliminaries

Recall that the category of generalized left R-modules

RC = (mod −R, Ab)

consist of additive covariant functors from the category of the finitely presented right R-modules mod −R to the category of abelian groups Ab. In this section we give some properties of the category _{R}C used later. For more detailed information about the category _{R}C we refer the reader to [2] and here we, for the most part, shall adhere to this paper. All subcategories considered are supposed to be full.

We say that a subcategory S of an abelian category C is a Serre subcategory if for every short exact sequence

0 ⟶ X ⟶ Y ⟶ Z ⟶ 0

in C the object Y ∈ S if and only if X, Z ∈ S. A Serre subcategory S of a Grothendieck category C is localizing if it is closed under taking direct limits. Equivalently, the inclusion functor i : S → C admits a right adjoint t = t_{S} : C → S which takes every object X ∈ C to the maximal subobject t(X) of X belonging to S. The functor t one calls the torsion functor.

An object X of a Grothendieck category C is finitely generated if whenever there are subobjects X_{i} ⊆ X with i ∈ I satisfying X = ∑_{i∈I} X_{i}, then there is a finite subset J ⊂ I such that X = ∑_{i∈J} X_{i}. The subcategory of finitely generated objects is denoted by fg C. A finitely generated object X is called finitely presented if every epimorphism γ : Y → X with Y ∈ fg C has the finitely generaed kernel Ker γ. By fp C we denote the subcategory consisting of finitely presented objects. Finally, we refer to a finitely presented object X ∈ C as coherent if every finitely generated subobject of X is finitely presented. The corresponding subcategory of coherent objects will be denoted by coh C.

The category _{R}C is a locally coherent Grothendieck category, that is every object C ∈ _{R}C is a direct limit C = \underrightarrow{lim}_{I}C_{i} of coherent objects C_{i} ∈ coh_{R}C. Equivalently, the category coh_{R}C is abelian. Moreover, _{R}C has enough coherent projective generators {(M, −)}_{M∈mod −R}. Thus, every coherent object C ∈ coh_{R}C has a projective presentation

(N, −) ⟶ (M, −) ⟶ C ⟶ 0,

where M, N ∈ mod −R.

We say that M ∈ _{R}C is a coh_{R}C-injective object if Ext_{_{R}C}^{1}(C, M) = 0 for every C ∈ coh_{R}C. The fully faithful functor −⊗_{R}? : R − Mod → _{R}C, M ↦ − ⊗_{R} M, identifies the module category R − Mod with the subcategory of coh_{R}C-injective objects of the category _{R}C. In addition, the functor FP-injective and weakly quasi-Frobenius rings 3

− ⊗_{R} M ∈ coh_{R}C if and only if M ∈ R − mod. Furthermore, for every C ∈ coh_{R}C there is also an exact sequence

0 ⟶ C ⟶ − ⊗_{R} M ⟶ − ⊗_{R} N

in coh_{R}C with M, N ∈ R − mod.

Proposition 1.1 ([3, 4]). For a ring R the following are equivalent: (1) R is left coherent;

(2) for every finitely presented right R-module M the left R-module M^{∗} = Hom_{R}(M, R) is finitely presented;

(3) for every finitely presented right R-module M the left R-module M^{∗} = Hom_{R}(M, R) is finitely generated;

(4) for every coherent object C ∈ coh_{R}C the left R-module C(R) is finitely presented;

(5) for every coherent object C ∈ coh_{R}C the left R-module C(R) is finitely generated.

Recall also that a monomorphism μ : M → N in R − Mod is a pure monomorphim if for every K ∈ Mod −R the morphism K ⊗ μ is a monomorphism. Equivalently, the _{R}C-morphism − ⊗ μ is a monomorphism.

In the sequel, we use the following Serre subcategories of the category coh_{R}C:

S^{R} = {C ∈ coh_{R}C | C(R) = 0}
S_{R} = {C ∈ coh_{R}C | (C, − ⊗_{R} R) = 0},

as well as the localizing subcategories S⃗^{R} and S⃗_{R} of _{R}C

S⃗^{R} = \{C ∈ _{R}C | C = \underrightarrow{lim} C_{i}, C_{i} ∈ S^{R}\}
S⃗_{R} = \{C ∈ _{R}C | C = \underrightarrow{lim} C_{i}, C_{i} ∈ S_{R}\}.

The corresponding S⃗^{R}-torsion and S⃗_{R}-torsion functors will be denoted by t_{S^{R}} and t_{S_{R}}.

2. F P -injective and weakly quasi-Frobenius rings

A left R-module M is said to be F P -injective (or absolutely pure) if for every F ∈ R − mod we have: Ext_{R}^{1}(F, M) = 0, or equivalently, every monomorphism μ : M → N is pure [5, 2.6]. The ring R is left F P -injective if the module _{R}R is F P -injective. M is an fp-injective module if for every monomorphism μ : K → L in R − mod the morphism (μ, M) is an epimorphism. Clearly, F P -injective modules are fp-injective and every finitely presented fp-injective module is F P -injective. M is called fp-flat if for every monomorphism μ : K → L in mod −R the morphism μ ⊗ M is a monomorphism.

We refer to a left R-module K as an F P -cogenerator if for every non-zero homomorphism f : M → N from the finitely generated module M to the finitely presented module N there exists g ∈ Hom_{R}(N, K) such that gf ≠ 0. K is said to be an fp-cogenerator if for every non-zero homomorphism f : M → N in R − mod there exists g ∈ Hom_{R}(N, K) such that gf ≠ 0.

4 Grigory Garkusha

Obviosly, F P -cogenerators are fp-cogenerators. On the other hand, it is not hard to see that any fp-cogenerator is an F P -cogenerator when the ring R is left coherent.

Lemma 2.1. For a left R-module K the following assertions are equivalent: (1) K is an F P -cogenerator;

(2) every finitely presented left R-module embeds in a product K^{I} = ∏_{I} K of copies of the module K;

(3) for every finitely presented left R-module M the following relation holds:

⋂_{φ∈Hom_{R}(M,K)} Ker φ = 0.

Proof. (1) ⇒ (2). Let M ∈ R−mod, I = Hom_{R}(M, K), and let μ : M → K^{I} be the homomorphism such that μ(x) = (φ(x))_{φ∈I}. If 0 ≠ x ∈ M and i : Rx → M is an inclusion, there is φ : M → K such that φi(x) ≠ 0. Thus, μ is a monomorphism.

(2) ⇒ (1). Let 0 ≠ f : M → N be a homomorphism from the finitely generated module M to the finitely presented module N. By assumption, there exists a monomophism g = (g_{i})_{i∈I} : N → K^{I}. Then gf ≠ 0, and so there is i_{0} ∈ I such that g_{i_{0}}f ≠ 0.

(1) ⇒ (3). Suppose μ : M → K^{I} is the monomorphism constructed in the proof of the implication (1) ⇒ (2). Then

⋂_{φ∈Hom_{R}(M,K)} Ker φ = Ker μ = 0.

(3) ⇒ (2). We may take I = Hom_{R}(M, K).

Recall that a module M ∈ R−Mod is semireflexive (respectively reflexive) if the canonical homomorphism M → M^{∗∗} is a monomorphism (respectively isomorphism).

We are now in possession of all the information for proving the following statement (cf. [1, 2.5]):

Theorem 2.2. For a ring R the following conditions are equivalent: (1) the module R_{R} is F P -injective;

(2) the module _{R}R is an F P -cogenerator;

(3) in R − Mod there is an fp-flat F P -cogenerator;

(4) in R − Mod there is an fp-flat cogenerator;

(5) if α : M → N is a morphism in R − mod such that α^{∗} = Hom_{R}(α, R) is an epimorphism, then α is a monomorphism;

(6) every finitely presented left R-module is semireflexive;

(7) every finitely presented left R-module embeds in an fp-flat module;

(8) every (injective) left R-module embeds in an fp-flat module;

(9) every F P -injective left R-module is fp-flat;

(10) every injective left R-module is fp-flat;

(11) every indecomposable injective left R-module is fp-flat;

(12) every flat right R-module is fp-injective.

FP-injective and weakly quasi-Frobenius rings 5

Proof. (1) ⇒ (2). According to [1, 2.5] S_{R} ⊆ S^{R} in _{R}C. Consider a non-zero homomorphism f : M → N with M ∈ fg(R − Mod) and N ∈ R − mod. Suppose C = Im(− ⊗ f); then C is a finitely generated subobject of the coherent object − ⊗_{R} N. Therefore C ∈ coh_{R}C. Assume that gf = 0 for every g ∈ N^{∗}. Consider an arbitrary _{R}C-morphism γ : C → − ⊗_{R} R. Since − ⊗_{R} R is a coh_{R}C-injective object, there exists − ⊗ h : − ⊗_{R} N → − ⊗_{R} R such that − ⊗ h|_{C} = γ. But hf = 0, and so γ = 0. Whence we obtain that C ∈ S_{R}, and thus C ∈ S^{R}. We see that C(R) = 0, which yields f = 0, a contradiction.

(2) ⇒ (3). By assumption, the module _{R}R is an F P -cogenerator.

(3) ⇒ (7). Since a direct product of fp-flat modules is an fp-flat module (see [1, 2.3]), our statement follows from lemma 2.1.

(7) ⇒ (1). Let f : M → F be an embedding of a module M ∈ R − mod in an fp-flat module F. Denote by X = Ker(− ⊗ f). Let X = ∑_{I} X_{i}, where each X_{i} ∈ fg_{R}C. Since X_{i} is a subobject of the coherent object − ⊗_{R} M, the object X_{i} is coherent itself. Because X(R) = 0, each X_{i} ∈ S^{R}. Consequently, X ∈ S⃗^{R}.

Let us apply now the left exact S_{R}-torsion functor t_{S_{R}} to the exact sequence

0 ⟶ X ⟶ − ⊗_{R} M ⟶ − ⊗_{R} F .

Since t_{S_{R}}(− ⊗_{R} F) = 0 (see [1, 2.2]), one gets that t_{S_{R}}(− ⊗_{R} M) = t_{S_{R}}(X) ⊆ X ∈ S⃗^{R}. Now let C ∈ S_{R}; since C is a subobject of − ⊗_{R} M for some M ∈ R − mod, from the relation

C = t_{S_{R}}(C) ⊆ t_{S_{R}}(− ⊗_{R} M) ∈ S⃗^{R}

we deduce that S_{R} ⊆ S^{R}, whence the ring R is right F P -injective by [1, 2.5].

(1) ⇒ (5). Let C = Ker(− ⊗ α); then C ∈ coh_{R}C and since − ⊗_{R} R is a coh_{R}C-injective object, there is an exact sequence of abelian groups

(− ⊗_{R} N, − ⊗_{R} R) ⟶^{(−⊗α,−⊗_{R}R)} (− ⊗_{R} M, − ⊗_{R} R) ⟶ (C, − ⊗_{R} R) ⟶ 0. (2.1)

Because α^{∗} is an epimorphism, we conclude that C ∈ S_{R} ⊆ S^{R}. Thus C(R) = 0, and hence α is a monomorphism.

(5) ⇒ (1). Let C ∈ S_{R}; then there is an exact sequence

0 → C → − ⊗_{R} M →^{−⊗α} − ⊗_{R} N,

which induces an exact sequence of the form (2.1). Since (C, − ⊗_{R} R) = 0, α^{∗} is an epimorphism, and hence a monomorphism. So C ∈ S^{R}. By [1, 2.5] the module R_{R} is F P -injective.

(1) ⇔ (6). This follows from [7, 2.3].

(4) ⇒ (3). Obvious.

(1) ⇒ (9), (10), (11). This is a consequence of [1, 2.5].

(9) ⇒ (10) ⇒ (11). Straightforward.

(11) ⇒ (4). Suppose that {M_{i} | i ∈ I} is a system of representatives for isomorphism classes of simple R-modules. Then every E_{i} = E(M_{i}) is 6 Grigory Garkusha

an indecomposable injective R-module and the module E = ∏_{I} E(M_{i}) is a cogenerator in R − Mod. Because every module E_{i} is fp-flat, the module E is fp-flat by [1, 2.3].

(8) ⇒ (7). Easy.

(10) ⇒ (8). It suffices to observe that the injective hull E(M) of the module M is an fp-flat module.

(1) ⇒ (12). This is a consequence of [1, 2.5].

(12) ⇒ (1). Since the module R_{R} is flat, it is fp-injective, and so is F P -injective.

A left (respectively right) ideal I of the ring R is annulet if I = 𝔩(X) (respectively I = 𝔯(X)), where X is some subset of the ring R and 𝔩(X) = {r ∈ R | rX = 0} (respectively 𝔯(X) = {r ∈ R | Xr = 0}). According to [8] the left ideal I is annulet if and only if I = 𝔩𝔯(I).

Proposition 2.3. For a ring R the following assertions hold:

(1) if R_{R} is an F P -injective module, then

(a) for arbitrary finitely generated right ideals I, J of the ring R one has: 𝔩(I ∩ J) = 𝔩(I) + 𝔩(J);

(b) for an arbitrary finitely generated left ideal I one has: I = 𝔩𝔯(I).

(2) if for R the conditions (a) and (b) from (1) hold, then every homomorphism of a finitely generated right ideal of the ring R into R can be extended to R.

Proof. The proof is similar to that of [9, 12.4.2].

Corollary 2.4. Under the conditions (a) and (b) of proposition 2.3 if the ring R is right coherent, then it is a right F P -injective ring.

Proof. By proposition 2.3 we have Ext_{R}^{1}(R/I, R) = 0 for every finitely generated right ideal I of the ring R. From [5, 3.1] we deduce that Ext_{R}^{1}(M, R) = 0 for every M ∈ mod −R, i.e. R is right F P -injective.

Now let us consider the situation when every finitely presented left module embeds in a free module. Rings with such a property one calls left IF -rings. In view of theorem 2.2 any left IF -ring is right F P -injective. The next statement extends the list of properties characterizing the IF -rings (cf. [3]).

Proposition 2.5. For a ring R the following conditions are equivalent: (1) R is a left IF -ring;

(2) every finitely presented left R-module embeds in a flat R-module;

(3) every F P -injective left R-module is flat;

(4) every injective left R-module is flat.

Proof. (1) ⇒ (2) is trivial.

(2) ⇒ (1). Let f : K → M be an embedding of a module K ∈ R − mod in a flat module M. By theorem of Govorov and Lazard [8, 11.32] the module M is a direct limit \underrightarrow{lim}_{I}P_{i} of the projective modules P_{i}. By [6, V.3.4] there FP-injective and weakly quasi-Frobenius rings 7

is i_{0} ∈ I such that f factors through P_{i_{0}}. Therefore K is a submodule of P_{i_{0}}. It remains to observe that P_{i_{0}} is a submodule of some free module.

(1) ⇔ (4). This follows from [3, 2.1].

(3) ⇒ (4). Obvious.

(4) ⇒ (3). If M is an F P -injective left R-module, then the sequence

0 ⟶ M ⟶ E ⟶ E/M ⟶ 0,

in which E = E(M), is pure. Since E is a flat module, the module E/M is flat by [6, I.11.1]. Let M̂ = Hom_{ℤ}(M, ℚ/ℤ) denote the character module of M. By [6, I.10.5] the modules Ê and { \atop E/M} are injective, and so the exact sequence

0 ⟶ { \atop E/M} ⟶ Ê ⟶ M̂ ⟶ 0

splits. Consequently, the module M̂ is injective, and so M is flat by [6, I.10.5].

Colby [3] has constructed an example of a left IF -ring, which is not a right IF -ring.

Proposition 2.6. If R is a left F P -injective ring and a left IF -ring, then it is right coherent.

Proof. By assumption, every K ∈ R − mod embeds in a free module (and so in a finitely generated free module as well). One has the following exact sequence

0 ⟶ K ⟶ R^{n} ⟶ R^{m}  (2.2)

in R−mod. Since the module_{R}R is F P -injective, one gets an exact sequence

R^{m} ⟶ R^{n} ⟶ K^{∗} ⟶ 0  (2.3)

in mod −R, hence K^{∗} ∈ mod −R. By proposition 1.1 the ring R is right coherent.

IF-problem. Is it true that any left IF -ring is right coherent?

It should be remarked that IF -problem, in view of proposition 2.5, is equivalent to Jain’s problem [7, p. 442]: will be the ring R right coherent if every injective left R-module is flat?

We recall that the ring R is almost regular (see [1]) if every (both left and right) module is fp-flat. By theorem 2.2 almost regular rings are two-sided F P -injective rings.

Corollary 2.7. An almost regular ring R will be a left IF -ring if and only if it is regular.

Proof. Clearly, an almost regular ring is regular if and only if it is left or right coherent. Therefore our assertion immediately follows from proposition 2.6.

8 Grigory Garkusha

Recall also that the ring R is indiscrete if it is a simple almost regular ring. Prest, Rothmaler and Ziegler [10] have constructed an example of a non-regular indiscrete ring.

The ring R is said to be weakly quasi-Frobenius (or WQF-ring) if it determines an R-duality between the categories of finitely presented left and right R-modules. Such rings can be described as (left and right) F P -injective (left and right) coherent rings [1, 2.11].

The next theorem extends the list of properties characterizing the WQFrings (cf. [1, 2.11; 2.12], [3, 2.2]).

Theorem 2.8. For a ring R the following assertions are equivalent: (1) R is a WQF-ring;

(2) R is a left and right IF -ring;

(3) R is left coherent and left F P -injective, and every cyclic finitely presented left R-module embeds in a free module;

(4) every left and every right F P -injective R-module is flat;

(5) every left and every right injective R-module is flat.

Proof. (1) ⇒ (4). This follows from [1, 2.11].

(2) ⇔ (4) ⇔ (5). Apply proposition 2.5.

(2) ⇒ (1). Since any left and right IF -ring is a two-sided F P -injective ring, our statement follows from proposition 2.6.

(1) ⇒ (3). Straightforward.

(3) ⇒ (1). Since for every cyclic K ∈ R − mod the dual module K^{∗} ≠ 0, the proof of right F P -injectivity of the ring R is similar to that of [1, 2.9].

Let us show that the ring R is right coherent. In view of proposition 1.1 it suffices to prove that K^{∗} ∈ mod −R for every K ∈ R − mod. We use induction on the number of generators n of the module K. When n = 1, considerating exact sequences (2.2) and (2.3) for K, one gets K^{∗} ∈ mod −R. If K is finitely presented on n generators, let K′ be the submodule of K generated by one of these generators. Since R is left coherent, the modules K′ and K/K′ are finitely presented on less than n generators. Because R is left F P -injective, one has an exact sequence

0 ⟶ (K/K′)^{∗} ⟶ K^{∗} ⟶ (K′)^{∗} ⟶ 0,

where both (K′)^{∗} and (K/K′)^{∗} are finitely presented by induction. Thus K^{∗} ∈ mod −R.

Now we combine the preceding arguments in the following theorem (cf. properties of QF-rings [11, 24.4], [9, 13.2.1] and also properties of rings with full duality [9, 12.1.1]):

Theorem 2.9. For a two-sided coherent ring R the following conditions are equivalent:

(1) R is a WQF-ring;

(2) the modules _{R}R and R_{R} are F P -injective;

(3) _{R}R and R_{R} are F P -cogenerators;

(4) the module R_{R} is an F P -injective F P -cogenerator; FP-injective and weakly quasi-Frobenius rings 9

(5) every left and every right finitely presented R-module is reflexive;

(6) every left and every right cyclic finitely presented R-module is reflexive;

(7) every left and every right cyclic finitely presented R-module embeds in a free module;

(8) for a finitely generated left ideal I and for a finitely generated right ideal J of the ring R one has: 𝔩𝔯(I) = I and 𝔯𝔩(J) = J.

Proof. (1) ⇔ (2). This follows from [1, 2.11].

(2) ⇔ (3) ⇔ (4). Apply theorem 2.2.

(2) ⇔ (5). This follows from [5, 4.9].

(5) ⇒ (6). Obvious.

(6) ⇒ (7). Let M be a cyclic finitely presented left R-module. In view of proposition 1.1 the module M^{∗} ∈ mod −R, and so there is an epimorphism R^{n} → M^{∗}. Hence M ≈ M^{∗∗} → R^{n} is a monomorphism.

(7) ⇒ (8). Let I be a finitely generated left ideal of the ring R. By assumption, the module R/I embeds in a free module. By [11, 20.26] there exists a finite subset X of R such that I = 𝔩(X), i. e., I is an annulet ideal. By symmetry, every finitely generated right ideal is annulet.

(8) ⇒ (2). In view of corollary 2.4 it suffices to show that for arbitrary finitely generated right ideals I and J of the ring R the following equality holds: 𝔩(I ∩ J) = 𝔩(I) + 𝔩(J). Since R is coherent by assumption, by the Chase theorem [6, I.13.3] both I ∩ J and 𝔩(I) + 𝔩(J) are finitely generated ideals.

One has

𝔯𝔩(I ∩ J) = I ∩ J = 𝔯𝔩(I) ∩ 𝔯𝔩(J) = 𝔯(𝔩(I) + 𝔩(J)).

Applying 𝔩, one gets

𝔩(I ∩ J) = 𝔩𝔯(𝔩(I) + 𝔩(J)) = 𝔩(I) + 𝔩(J).

Thus R_{R} is F P -injective. Likewise, _{R}R is F P -injective.

It is well-known that QF-rings have the global dimension to be equal to 0 (and then the ring R is semisimple), or ∞. In turn, WQF-rings, in view of [5, 3.6], have the weak global dimension to be equal to 0 (and then the ring R is regular), or ∞.

Some examples of WQF-rings the reader can find in [3].

Next, we consider the class of rings over which every finitely generated left R-module embeds in a free R-module. Such rings we shall call left FGFrings. Clearly, any FGF-ring will be an IF -ring. In turn, if every cyclic left R-module embeds in a free R-module, the ring R one calls a left F-ring. The following problems are still open (see [12]):

FGF-problem. Does the class of left FGF-rings coincide with the class of QF-rings?

CF-problem. Will be left CF-rings left artinian?

Recall also that the ring R is a left Kasch ring if the injective hull E(_{R}R) of the module _{R}R is an injective cogenerator in R − Mod. Equivalently, for 10 Grigory Garkusha

every non-zero cyclic left R-module M the dual module M^{∗} ≠ 0 (see [6, XI.5.1]).

Lemma 2.10. For a ring R the following assertions hold:

(1) if R is a left coherent ring and a left CF-ring, then it is a left noetherian ring and a left Kasch ring;

(2) if R is a left noetherian ring and a left IF -ring, then it is a left FGFring. If R is a left coherent and a left FGF-ring, then it is a left noetherian ring and a left IF -ring.

Proof. (1). It is easy to see that R is a left Kasch ring. Thus we must show that the module _{R}R is noetherian.

Suppose I is a left ideal of the ring R. By assumption, the module R/I is a submodule of a free R-module R^{n} for some n ∈ ℕ. Since the ring R is left coherent, the module R^{n} is coherent, and hence the module R/I is finitely presented, i.e., I is a finitely generated ideal.

(2). It is necessary to observe that over a noetherian ring every finitely generated module is finitely presented and also make use of the first statement.

Proposition 2.11. For a two-sided coherent ring R the following assertions are equivalent:

(1) R is a left FGF-ring;

(2) the module _{R}R is a noetherian F P -cogenerator;

(3) the module _{R}R is noetherian and the module R_{R} is F P -injective;

(4) R is a left noetherian ring, a left Kasch ring, and the module E(_{R}R) is flat.

Proof. (1) ⇒ (2), (3). This follows from lemma 2.10, lemma 2.1 and theorem 2.2.

(1) ⇒ (4). Apply lemma 2.10 and proposition 2.5.

(2) ⇔ (3). This is a consequence of theorem 2.2.

(2) ⇒ (1). Since R is a left noetherian ring, every finitely generated left R-module is finitely presented. By lemma 2.1 every finitely presented left R-module is a submodule of the module R^{I}. Because the ring R is right coherent, the module R^{I} is flat by [6, I.13.3]. By proposition 2.5 R is a left IF -ring and by lemma 2.10 R is also a left FGF-ring.

(4) ⇒ (1). In this case the proof is similar to the proof of the implication (2) ⇒ (1) if we observe that every finitely generated left R-module is a submodule of the flat module E^{I}.

The ring R is called left semiartinian if every non-zero cyclic left R-module has a non-zero socle. R is semiregular if R/ rad R is a regular ring.

Lemma 2.12. [12, 2.1] A left CF-ring R is left semiartinian if and only if soc(_{R}R) is an essential submodule in _{R}R.

The next two statements partially solve CF- and FGF-problems respectively (cf. [12]):

FP-injective and weakly quasi-Frobenius rings 11

Proposition 2.13. Suppose R is a left noetherian ring and a left CF-ring. Then the following conditions are equivalent for R:

(1) R is left artinian;

(2) R is left or right semiartinian;

(3) R is a semiregular ring;

(4) R is a semiperfect ring;

(5) soc(_{R}R) is an essential submodule in _{R}R.

Proof. (1) ⇒ (2). Any left artinian ring is left perfect, and so is right semiartinian by [6, VIII.5.1].

(2) ⇒ (1). Our assertion follows from [6, VIII.5.2].

(1) ⇒ (3). Easy.

(3) ⇒ (4). Since soc(_{R}R) is a finitely generated left ideal of R, our assertion follows from [12, 4.1].

(4) ⇒ (5). Obvious.

(5) ⇒ (1). By the preceding lemma R is left semiartinian and since R is left noetherian, our assertion follows from [6, VIII.5.2].

Proposition 2.14. Let R be a left noetherian ring and a left FGF-ring. Then the following are equivalent for R:

(1) R is a QF-ring;

(2) R is a WQF-ring;

(3) R is a left F P -injective ring;

(4) R is a right Kasch ring;

(5) R is a semiregular ring;

(6) R is left or right semiartinian;

(7) soc(_{R}R) is an essential submodule in _{R}R.

Proof. The implications (1) ⇒ (2) ⇒ (3), (1) ⇒ (4), (5) are obvious. (3) ⇒ (1). Our assertion follows from [1, 2.6].

(4) ⇒ (3). Over a right Kasch ring the module M^{∗} ≠ 0 for every non-zero cyclic right R-module M. Since R is a left FGF-ring, by theorem 2.2 R is a right F P -injective ring. Therefore R is a left F P -injective ring by [1, 2.9].

(5) ⇔ (6) ⇔ (7). Apply proposition 2.13.

(7) ⇒ (1). By proposition 2.13 the ring R is left artinian. Since any left artinian ring is right perfect, our assertion follows from [12, 2.5].

3. Group rings

Let R be a ring and G a group. Denote the group ring of G with coefficients in R by R(G).

Lemma 3.1. Suppose G is a finite group and M ∈ Mod −R(G). Let f = (f_{i})_{I} : M → R^{I} is some R-monomorphism with I some set of indices. Then the R(G)-homomorphism f̃ = (f̃_{i})_{I} : M → R(G)^{I} defined by the rule

f̃_{i}(m) = ∑_{g∈G} f_{i}(mg)g^{−1}

is an R(G)-monomorphism.

Proof. It is directly verified that f̃ is indeed a monomorphism of R(G)- modules.

According to Renault’s theorem [13] the group ring R(G) is left selfinjective if and only if the ring R is left self-injective and the group G is finite. In turn, for the F P -injective rings there is the following statement.

Theorem 3.2. The group ring R(G) is left F P -injective if and only if the ring R is left F P -injective and the group G is locally finite.

Proof. Suppose R(G) is left F P -injective and M ∈ mod −R. Then the module M ⊗_{R} R(G) ∈ mod −R(G) and, in view of lemma 2.1 and theorem 2.2, there exists a monomorphism μ : M ⊗_{R} R(G) → R(G)^{I} with I some set of indices. Then the composition

M ⟶^{β} M ⊗_{R} R(G) ⟶^{μ} R(G)^{I}

of morphisms μ and β is an R-monomorphism, where β(m) = m ⊗ e, e is a unit of the group G. Since R(G) is a free R-module, the R-module R(G)^{I} is fp-flat by [1, 2.3]. Theorem 2.2 implies that the ring R is left F P -injective.

Let us show now that the group G is locally finite. Let H be a non-trivial subgroup of the group G generated by elements {h_{i}}_{i=1}^{n}. Then the right ideal ωH of the ring R(G) generated by the elements {1 − h_{i}}_{i=1}^{n} is non-zero. By proposition 2.3 𝔩(ωH) ≠ 0, and so H is finite by [14, 2.1].

Now let R be left F P -injective and the group G locally finite. To begin, let us show that the ring R(G) is left F P -injective if G is finite.

Suppose M ∈ mod −R(G). Because the group G is finite, M ∈ mod −R. Since the module R_{R} is an F P -cogenerator by theorem 2.2, by lemma 2.1 M_{R} is a submodule of R^{I}, where I is some set. By lemma 3.1 M_{R(G)} is a submodule of R(G)^{I}. Consequently, R(G) is an F P -cogenerator, and hence R(G) is a left F P -injective ring by theorem 2.2.

Next, suppose that G is an arbitrary locally finite group and M ∈ R(G) − mod. Then there is a short exact sequence of R(G)-modules

0 ⟶ K ⟶^{i} R(G)^{n} ⟶^{p} M ⟶ 0.

Let X be a finite set of generators for _{R(G)}K. Because G is locally finite, there is a finite subgroup H of G such that R(H)X ⊆ R(H)^{n} ⊆ R(G)^{n}. We result in the short exact sequence of R(H)-modules

0 ⟶ K′ ⟶^{ī} R(H)^{n} ⟶^{p̄} M′ ⟶ 0,  (3.1)

where K′ = R(H)X, M′ = R(H)Y, Y is a finite set of generators of the module_{R(G)}M. Tensoring sequence (3.1) on R(G)_{R(H)}, one gets the following commutative diagram with exact rows:

0 ⟶ R(G) ⨂_{R(H)} K′ ⟶^{1⊗ī} R(G) ⨂_{R(H)} R(H)^{n} ⟶^{1⊗p̄} R(G) ⨂_{R(H)} M′ ⟶ 0
α↓  ↓^{β}  ↓^{γ}
0 ⟶  K  ⟶_{i}  R(G)^{n}  ⟶_{p}  M  ⟶ 0,

in which β is an isomorphism, α(r ⊗ x) = rx. Clearly, α is an isomorphism, and hence γ is an isomorphism.

If we showed that every f ∈ Hom_{R(G)}(K, R(G)) is extended to some σ ∈ Hom_{R(G)}(R(G)^{n}, R(G)), we would obtain that Ext_{R(G)}^{1}(M, R(G)) = 0, as required.

So suppose that f ∈ Hom_{R(G)}(K, R(G)) and τ ∈ Hom_{R(H)}(R(G), R(H)), τ(∑_{g∈G} r(g)g) = ∑_{g∈H} r(g)g. Consider f̄ = τf|_{K′} ∈ Hom_{R(H)}(K′, R(H)). Because R(H) is a left F P -injective ring, there is σ̄ : R(H)^{n} → R(H) such that f̄ = σ̄ī. One gets

f = (1 ⊗ f̄)α^{−1} = (1 ⊗ σ̄)(1 ⊗ ī)α^{−1} = (1 ⊗ σ̄)β^{−1}i = σi,

where σ = (1 ⊗ σ̄)β^{−1} is the required homomorphism.

The Renault theorem and theorem 3.2 implies that given an arbitrary self-injective ring R, one can construct F P -injective rings which are not selfinjective. To be definite, the following statement holds:

Corollary 3.3. If R is a left self-injective ring, G is a locally finite group, and |G| = ∞, then the group ring R(G) is left F P -injective but not left self-injective.

Proposition 3.4 (Colby [3]). The group ring R(G) is a left IF -ring if and only if R is a left IF -ring and the group G is locally finite.

Proof. If R(G) is a left IF -ring, it is also right F P -injective by theorem 2.2. By the preceding theorem the group G is locally finite. Similar to the proof of theorem 3.2, given M ∈ R−mod there is a composition of R-monomorphisms

M ⟶^{β} R(G) ⊗_{R} M ⟶^{μ} R(G)^{n}

with β(m) = e ⊗ m. Since R(G) is a free R-module, the module M is a submodule of the free R-module R(G)^{n}.

Conversely, let R be a left IF -ring and G a locally finite group. First, let us prove that R(G) is a left IF -ring if G is a finite group. For this consider M ∈ R(G)−mod. Since G is finite, M ∈ R−mod, and so_{R}M is a submodule of R^{n} for some n ∈ ℕ. By lemma 3.1 _{R(G)}M is a submodule of R(G)^{n}, i.., R(G) is indeed a left IF -ring.

If G is an arbitrary locally finite group, then for any M ∈ R(G) − mod there is a finite subgroup H of the group G such that

M ≈ R(G) ⊗_{R(H)} R(H)Y

14 Grigory Garkusha

with Y a finite set of generators for _{R(G)}M (see the proof of theorem 3.2). By assumption, the R(H)-module R(H)Y is a submodule of a free module R(H)^{n} for some n ∈ ℕ. Thus, M is a submodule of the free module R(G)^{n} ≈ R(G) ⊗_{R(H)} R(H)^{n}.

Theorem 3.5. The group ring R(G) is weakly quasi-Frobenius if and only if the ring R is weakly quasi-Frobenius and the group G is locally finite.

Proof. Theorem 2.8 implies that any WQF-ring is a left and right IF -ring. Therefore our statement immediately follows from proposition 3.4.

It is well-known that the group ring R(G) is semisimple (see [14, 8]) if and only if the ring R is semisimple, the group G is finite, and |G| is invertible in R. In turn, by theorem of Auslander and McLaughlin (see [14]) R(G) is a regular ring if and only if the ring R is regular, the group G is locally finite, and for every finite subgroup H of G the equality |H| = n implies nR = R.

To conclude, we give some examples of WQF-rings which are simultaneously neither QF-rings, nor regular rings.

Examples. (1) Given an arbitrary regular ring R, we can construct WQFrings which will not be regular. Namely, it is necessary to consider an arbitrary locally finite group G, in which there is at least one finite subgroup H of G such that the order |H| is not a unit in R.

To take an example, consider the field K of the characteristic p ≠ 0. Let R = ∏_{i=1}^{∞} K_{i}, K_{i} = K, be the ring with component-wise operations. Then R is a regular but not semisimple ring, as one easily sees. If G is a finite group such that p devides |G|, then the ring R(G) is a weakly quasi-Frobenius ring being neither quasi-Frobenius, nor regular.

(2) Let R be an arbitrary QF-ring, G an arbitrary locally finite group, and |G| = ∞. Then R(G) is a weakly quasi-Frobenius ring but not quasi-Frobenius. Moreover, R(G) is regular if and only if R is a semisimple ring and the order of every finite subgroup of G is invertible in R.

As an example, if K is the field of the characteristic p ≠ 0, the group G = ∪_{k≥1}G_{k}, where every G_{k} is a cyclic group with a generator a_{k} of the order p^{k}, and a_{k} = a_{k+1}^{p}, then the group algebra K(G) = \underrightarrow{lim}_{k}K(G_{k}) is weakly quasi-Frobenius (see also [15]) being neither quasi-Frobenius, nor regular.

References

[1] G. A. Garkusha, A. I. Generalov, Duality for categories of finitely presented modules, Algebra i Analiz 11 (1999), issue 6, 139-153 (in Russian). English transl. in St. Petersburg Math. J. 11 (2000), No. 6.

[2] I. Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. 74 (1997), 503-558.

[3] R. R. Colby, Rings which have flat injective modules, J. Algebra 35 (1975), 239-252.

[4] G. Garkusha, Grothendieck categories, Preprint (http://xxx.lanl.gov), 1999.

[5] B. Stenström, Coherent rings and F P -injective modules, J. London Math. Soc. 2 (1970), 323-329.

FP-injective and weakly quasi-Frobenius rings 15

[6] B. Stenström, Rings of quotients, Grundlehren Math. Wiss., vol. 217, Springer-Verlag, New York–Heidelberg, 1975.

[7] S. Jain, Flat and F P -injectivity, Proc. Amer. Math. Soc. 41 (1973), 437-442.

[8] C. Faith, Algebra: rings, modules and categories. I, Grundlehren Math. Wiss., vol. 190, Springer-Verlag, New York–Heidelberg, 1973.

[9] F. Kasch, Moduln und Ringe, B. G . Teubner, Stuttgart, 1977.

[10] M. Prest, Ph. Rothmaler, M. Ziegler, Absolutely pure and flat modules and “indiscrete” rings, J. Algebra 174 (1995), 349-372.

[11] C. Faith, Algebra. II. Ring theory, Grundlehren Math. Wiss., vol. 191, Springer-Verlag, Berlin–New York, 1976.

[12] J. Rada, M. Saorin, On two problems about embedding of modules in free modules, Notas de Matemática (Serie: Pre-Print) 170 (1998).

[13] G. Renault, Sur les anneaux de groupes, C. R. Acad. Sc. Paris Ser. A 273 No. 2 (1971), 84-87.

[14] I. G. Connell, On the group ring, Canad. J. Math. 15 (1963), 650-685.

[15] M.-P. Malliavin, Sur les anneaux de groupes F P -self-injectifs, C. R. Acad. Sc. Paris Ser. A 273 No. 2 (1971), 88-91.

Stary Peterhof, Russia

E-mail address: ggarkusha@mail.ru