REFLECTIVE AND COREFLECTIVE SUBCATEGORIES

JUAN RADA, MANUEL SAORÍN, AND ALBERTO DEL VALLE

Dedicated to Kent Fuller on his 60th anniversary

Abstract. A full subcategory F of a category A is reflective (resp. coreflective) if and only if every object of A has a F-envelope (resp. a F-cover) which completes the associated diagrams in a unique way. Using this viewpoint, we give several charecterizations and examples of reflective and coreflective subcategories, mainly in abelian and module categories.

1. Introduction

In recent times, there has been strong interest in the study of (pre)envelopes and (pre)covers by an arbitrary class F of modules (cf. [11],[25],[2],[26]). One of the open problems there is to find when every module has a F-envelope completing diagrams in a unique way. Results for the case when F is the class of projective or flat modules can be found in [2],[10],[12]. For an arbitrary class F of objects of a category A, the existence of a F-envelope completing diagrams uniquely for every object of A is equivalent to the fact that F is a reflective subcategory of Mod_{R} in the classical terminology of category theory (see, e.g., [31]). The goal of this work is twofold. On one side, we shall show that Freyd’s adjoint functor theorem, in the case of an inclusion functor, is closely related to the notion of covariantly finite subcategory introduced by Auslander’s school, which also allows us in this case to extend Freyd’s theorem to noncomplete categories (see Theorem 3.1 and its corollaries). By using that, as a second objective, we shall give a good supply of examples of reflective and coreflective subcategories, mainly in module and abelian categories.

After introducing our notation and terminology, Section 3 contains the already mentioned generalizations of Freyd’s theorem which allow applications to non-complete categories; also, given a locally finitely presented abelian category, we give criteria for a subcategory consisting of direct limits of its finitely presented objects to be reflective. In Section 4 we give applications of the previous results, with special attention to subcategories of an abelian category A of the form Copres (M) for an object M of A. Finally, Section 5 studies the reflective subcategories of Mod_{R} which are generalized Krull-Schmidt (see definition in Section 5).

1991 Mathematics Subject Classification. 18A40, 16D90, 16E70.

The second and third authors are supported by the DGICYT of Spain (PB93-0515-C02-02) and by the Comunidad Autónoma de la Región de Murcia (PIB94/25).

2. Notation and terminology

Let A be any category and F any class of objects of A (all classes of objects in a category are considered to be closed under isomorphisms, and we shall usually identify F with the full subcategory of A whose objects are those of F). An object of F will be called an F-object, and a F-morphism will be a morphism between F-objects.

A F-preenvelope of the A-object A is a morphism λ : A → F with F ∈ F such that, for any other morphism λ′ : A → F′ with F′ ∈ F, there exists a morphism μ : F → F′ such that μ ∘ λ = λ′. This means that the map A(F, F′) → A(A, F′) induced by λ is surjective for any F-object F′. A F-preenvelope of A is also called a left F-approximation of A, and the class F is said to be covariantly finite when every A-object has a F-preenvelope. The F-preenvelope λ : A → F is said to be a F-envelope (or a minimal left F-approximation) when it is left minimal, i.e. when the preimages of λ via A(F, F) → A(A, F) are all automorphism of F. The dual concepts are F-precover (or right F-approximation), contravariantly finite class and F-cover (or minimal right F-approximation).

A F-(pre)envelope A → F for which the induced maps A(F, F′) → A(M, F′) (with F′ ∈ F) are all bijective is called a F-reflection of A ([18],[24]; it is called a F-envelope which complete diagrams in a unique way in [2]). A full subcategory F of A is said to be reflective in A ([18],[24]) when the inclusion functor U : F ↪ A has a left adjoint F : A → F (called a reflector for F). In this case the unit of the adjunction η : 1_{A} → U∘F gives a F-reflection η_{A} for each A-object A and, conversely, when each A-object has a Freflection, it is clear how to define a reflector for F. The dual concepts are F-coreflection and coreflective subcategory.

The class F is said to be locally initially small in A [26] if, for every A-object A, there exists a set F_{A} ⊆ F such that every morphism A → F with F ∈ F factors through a direct product of modules in F_{A}. The dual concept is that of a locally finally small class. The class F is said to be closed for retracts (in A) if, whenever F ∈ F and F → F′ is a retraction in A, we have F′ ∈ F. It is easy to see that, if F is reflective in A (or, more generally, if every object of A has a F-envelope), then F is closed for retracts.

Most of our results are given for well-powered categories, i.e., categories in which the subobjects of every given object form a set. Following [9], an object X of a preadditive category A with direct limits will be called finitely presented when the functor Hom_{A} (X, −) preserves direct limits. When the finitely presented objects of A form a skeletally small class fp (A) of generators, we shall say that A is locally finitely presented. In that case, the finitely generated objects of A are just the homomorphic images of morphisms with finitely presented domain. When, moreover, every finitely generated subobject of a finitely presented one is again finitely presented, we shall say that A is locally coherent. For a class F of objects of fp (A) we shall denote by F⃗ the class of all objects of A that are (isomorphic to) direct limits of objects in F.

Following [30], a coperfect object in a locally finitely presented category is one that satisfies DCC on finitely generated subobjects. When every object of A is coperfect, we say that A is a coperfect category.

All our rings are associative with identity, and all our modules are unital. We denote by Mod_{R} (resp. mod_{R}) the category of all (resp. all finitely presented) right R-modules. We write pd (M_{R}) for the projective dimension of the right R-module M_{R}, and rgD (R) (resp. lgD (R), wD (R)) for the right global (resp. left global, weak global) dimension of the ring R.

3. Characterizations of reflective subcategories

Definition 3.1. Let A be any category and let F be a full subcategory of A. Then:

We say that (finite) products of F-objects exist in A when any (finite) family of F-objects has a product in A.

We say that equalizers of F-morphisms exist in A when any pair of F-morphisms with the same domain and codomain has an equalizer in A. If A has zero morphisms we get, as a particular case, the definition that kernels of F-morphisms exist in A.

When equalizers of F-morphisms exist in A and, for any F-object F, every downward directed family {Eq (f_{i}, g_{i})}_{i∈I} (indexed by a set) of equalizers of pairs of F-morphisms with domain F and common codomain has the property that ∩_{i∈I} Eq (f_{i}, g_{i}) exists in A and, moreover, ∩_{i∈I} Eq (f_{i}, g_{i}) = Eq (f, g) for some pair of F-morphism f, g with domain F and common codomain, then we shall say that strong equalizers of F-morphisms exist in A. Similarly, one gets the definition that strong kernels of F-morphisms exist in A.

Before giving our results, we present several examples of classes with the properties defined above.

Example 3.1. (1) Let A be a complete category and F be any class of objects in A. Then products of F-objects and equalizers of F-morphisms always exist in A.

(2) Let A be a locally coherent abelian category; then, for every class F of objects in fp (A), kernels of F-morphisms exist in fp (A). Moreover, if F has products in A, is closed for retracts and finite products and, for every morphism f : F → ∏_{i∈I} F_{i} with F and each F_{i} in F, there exists a finite subset J of I such that Ker f = Ker (π_{J} ∘ f), where π_{J} : ∏_{i∈I} F_{i} → ∏_{i∈J} F_{i} is the canonical projection, then strong kernels of F-morphisms exist in fp (A). In particular, this happens when F consists of coperfect objects.

(3) A particular case of (2) is the following: If R is a right coherent and left perfect ring, then for every class F ⊆ mod_{R} closed for direct summands and finite direct sums, strong kernels of F-morphisms exist in mod_{R}.

(4) If A is a locally finitely presented abelian category and F ⊆ fp (A) is a class of objects of finite length all whose composition factors are finitely presented, then strong kernels of F-morphisms exist in fp (A).

Theorem 3.1. Let A be a well-powered category and let F be a full subcategory of A such that equalizers of F-morphisms and intersections of them exist in A. The following assertions are equivalent:

(a) F is reflective in A.

(b) F is covariantly finite in A and closed for intersections of equalizers of F-morphisms.

Proof. (a)⇒(b). This follows by [18, (36.13)].

(b)⇒(a). Let λ : A → F be a F-preenvelope of an arbitrary A-object A and consider the class Ω of all pairs of morphisms f, g : F → F′ with F′ ∈ F and f ∘ λ = g ∘ λ. Since A is well-powered, we can fix a family {f_{i}, g_{i} : F → F_{i} | i ∈ I} (indexed by a set I) of elements of Ω such that the equalizer of every pair of morphisms in Ω equals Eq (f_{i}, g_{i}) for some i ∈ I. Let now e : E → F be the intersection of the family {Eq (f_{i}, g_{i}) | i ∈ I}; by hypothesis we have E ∈ F, and by construction there exists a unique morphism μ : A → E such that e ∘ μ = λ. Then it is clear that μ is a F-preenvelope of A, and we shall prove that it is indeed an F-reflection, which will finish the proof of the theorem. For this, we start by noting that, whenever f, g : F → F′ is a pair of F-morphisms with f ∘ λ = g ∘ λ and e′ : E′ → F is its equalizer, there is a unique monomorphism u : E → E′ such that e = e′ ∘ u. Now we check that e is a section: the preenveloping condition of λ gives a morphism p : F → E with μ = p ∘ λ, and hence

e ∘ p ∘ λ = e ∘ μ = λ = 1_{F} ∘ λ.

If e′ : E′ → F is the equalizer of 1_{F} and e ∘ p then there is a monomorphism u : E → E′ with e = e′ ∘ u, and therefore

e ∘ 1_{E} = e = 1_{F} ∘ e = 1_{F} ∘ e′ ∘ u = e ∘ p ∘ e′ ∘ u = e ∘ p ∘ e;

as e is monic this implies that p ∘ e = 1_{E}, as claimed. Now suppose that h, h′ : E → G are F-morphisms with h ∘ μ = h′ ∘ μ. Then h ∘ p ∘ λ = h′ ∘ p ∘ λ, and if e′′ : E′′ → F is the equalizer of h ∘ p and h′ ∘ p then there is a monomorphism v : E → E′′ with e = e′′ ∘ v, and therefore

h = h ∘ p ∘ e = h ∘ p ∘ e′′ ∘ v = h′ ∘ p ∘ e′′ ∘ v = h′ ∘ p ∘ e = h′. □

The following result generalizes Freyd’s adjoint functor theorem for inclusion functors (see [24, p.177, Theorem 2]). Note that the condition “F is locally initially small” is more general than the “solution set condition” of the above cited result; indeed, if R is any ring and F is the class of all flat right R-modules, then F is locally initially small in Mod_{R} [26, Proposition 2.8], but it has the solution set condition if and only if R is left coherent [11, Proposition 5.1].

Corollary 3.2. Let A be a well-powered category and let F be a full subcategory of A such that products of F-objects and equalizers of F-morphisms exist in A. The following assertions are equivalent:

(a) F is reflective in A.

(b) F is covariantly finite in A and closed for equalizers.

(c) F is locally initially small in A and closed for equalizers and products.

Proof. (a)⇒(b) follows as before, (b)⇔(c) is proven as in [26, Theorem 3.3] (note that F is closed for retracts since it is closed for equalizers), and (b,c)⇒(a) follows from [18, (18.23)] and Theorem 3.1. □

Next we give a third partial version of Theorem 3.1 that is useful to deal with noncomplete categories.

Corollary 3.3. Let A be a well-powered category and let F be a full subcategory of A such that finite products of F-objects and strong equalizers of F-morphisms exist in A. Then the following assertions are equivalent:

(a) F is reflective in A.

(b) F is covariantly finite in A and closed for equalizers.

Proof. We only need to prove (b)⇒(a), and for this one can adapt the proof of Theorem 3.1 after showing that, given {f_{i}, g_{i} : F → F_{i} | i ∈ I} as there, ∩_{i∈I} Eq (f_{i}, g_{i}) exists and belongs to F. Now, by the hypotheses, it is enough to show that {Eq (f_{i}, g_{i}) | i ∈ I} is a downward directed family. For, given i, j ∈ I, let F′ be the product of F_{i} and F_{j} and let f (resp. g) be the morphism F → F′ induced by f_{i}, f_{j} (resp. g_{i}, g_{j}). By hypothesis, there exists k ∈ I such that Eq (f, g) = Eq (f_{k}, g_{k}) , and this is a subobject of both Eq (f_{i}, g_{i}) and Eq (f_{j}, g_{j}) . □

Remark 3.1. (1) Corollary 3.2 applies to Example 3.1.(1), showing that a full subcategory of a well-powered complete category A is reflective in A if and only if it is locally initially small and closed for products and equalizers.

(2) Corollary 3.3 applies to Example 3.1.(2-4), showing for instance that if A is a locally coherent abelian category and F ⊆ fp (A) is a subcategory consisting of coperfect objects, then F is reflective in fp (A) if and only if F is covariantly finite in fp (A) and closed for kernels and finite direct sums.

Remark 3.2. The above results admit straightforward dualizations which are left to the reader.

For suitable classes F, the uniqueness in the completion of diagrams allows to construct, from a F-reflection for each member of a direct system, a F-reflection for the direct limit of the system. This fact is exploited in the following two results.

Proposition 3.4. Let A be a locally finitely presented category, and let F ⊆ fp (A) be a class of objects closed for retracts. Then F is reflective in fp (A) if and only if F⃗ is reflective in A.

Proof. (a)⇒(b). First note that, if X ∈ fp (A) and λ : X → F is a F-(pre)envelope, then it is a F⃗-(pre)envelope. Moreover, if it is a F-reflection then it is also a F⃗-reflection. To see this it is enough to show that, if F̄ ∈ F⃗ (say F̄ = \underrightarrow{lim}F_{i} with each F_{i} ∈ F and canonical maps ϕ_{i} : F_{i} → F̄), then any morphism α : F → F̄ with α ∘ λ = 0 must be zero. By [9, Lemma 1.1.3], there exists an index i and a morphism β : F → F_{i} with β ∘ λ = 0 such that α = ϕ_{i} ∘ β. Now, the fact that λ is a F-reflection gives β = 0 and hence α = 0. The rest of the proof follows that of [25, Theorem 2.11].

(b)⇒(a). If X ∈ fp (A) and λ : X → F is a F⃗-reflection, where F = \underrightarrow{lim}F_{i} with each F_{i} ∈ F and canonical maps ϕ_{i} : F_{i} → F, then λ = ϕ_{i} ∘ λ′ for some λ′ : X → F_{i}, but in turn this λ′ factors as λ′ = h ∘ λ and, by the uniqueness in the completion of diagrams, we get ϕ_{i} ∘ h = 1_{F}. So, by hypothesis, F ∈ F and therefore λ is a F-reflection. □

Remark 3.3. (1) We cannot weave the hypothesis that F is closed for retracts. For example, if R is a von Neumann regular ring and F is the class of all finitely generated free modules, then F⃗ = Mod_{R} but F is not reflective in mod_{R} unless R is a division ring.

(2) Putting together [9, Theorem 4.2], [22, Theorem 12.3] and results of [15, §7 & ff.], one easily deduces that, in the situation of our Proposition 3.4, if A is (co)complete and kernels of F-morphisms are finitely presented, then (a-b) hold if and only if F⃗ is closed for products and kernels, if and only if F⃗ is closed for products and F is closed for kernels.

(3) Proposition 3.4 together with the above remark can be used to charecterize coreflectivity of subcategories of mod_{R}, when R is a ring with self-duality [1, §30].

4. Applications

We start by applying the results in Section 3 to some “classical” classes of modules. We write Inj_{R}, Proj_{R}, Flat_{R}, and FPInj_{R} for the classes of all injective, projective, flat and FP-injective right R-modules, respectively. Also, P^{<∞}, I^{<∞} and F^{<∞} denote the classes of right R-modules of finite projective, injective and flat dimension, respectively.

Proposition 4.1. For any ring R, the following conditions hold:

(i) Inj_{R} is reflective in Mod_{R} iff Proj_{R} is coreflective in Mod_{R} iff R is semisimple. (ii) FPInj_{R} is reflective in Mod_{R} iff Flat_{R} is coreflective in Mod_{R} iff R is von Neu-

mann regular. (iii) P^{<∞} is reflective in Mod_{R} iff P^{<∞} is coreflective in Mod_{R} iff I^{<∞} is reflective

in Mod_{R} iff I^{<∞} is coreflective in Mod_{R} iff rgD (R) < ∞. (iv) F^{<∞} is reflective in Mod_{R} iff F^{<∞} is coreflective in Mod_{R} iff wD (R) < ∞.

(v) Flat_{R} is reflective in Mod_{R} iff R is left coherent and wD (R) ≤ 2. [2, Prop. 2.1] (vi) Proj_{R} is reflective in Mod_{R} iff R is right perfect, left coherent, and rgD (R) ≤ 2. (vii) Inj_{R} is coreflective in Mod_{R} iff R is right noetherian and rgD (R) ≤ 2.

Proof. (i-iv) follow from Corollary 3.2 and its dual, since each of the relevant classes equals Mod_{R} when it is closed for kernels or cokernels (depending on whether we are considering reflectivity or coreflectivity). By [26, Propositions 2.8 and 2.9], Flat_{R} and Proj_{R} are always locally initially small classes, so that (v) and (vi) follow again by Corollary 3.2. Finally, if R is right noetherian then Inj_{R} is a locally finally small class and therefore (vii) follows from the dual of Corollary 3.2. □

In the next application we consider a skeletally small preadditive category C, and write (C^{op}, Ab) for the category of contravariant functors from C to the category Ab of abelian groups. A functor F ∈ (C^{op}, Ab) is representable if it is naturally isomorphic to (−, C) = Hom_{C} (−, C) for some C ∈ C, and a direct limit of representable functors is said to be flat (see, e.g., [9]).

Proposition 4.2. Let C be as above, and let R (resp. F) be the class of all representable (resp. flat) functors in (C^{op}, Ab). Then the following assertions are equivalent:

(a) C has cokernels.

(b) R is a reflective subcategory of fp (C^{op}, Ab).

(c) F is a reflective subcategory of (C^{op}, Ab).

Proof. Assume that C has cokernels and let F ∈ fp (C^{op}, Ab); then there exists a morphism

f : C → C′ in C and an exact sequence (−, C) ⟶^{(−,f)} (−, C′) ⟶^{τ} F → 0 in (C^{op}, Ab). If g : C′ → C′′ is the cokernel of f then (−, g) factors through τ and, using Yoneda’s Lemma, it is easy to see that the morphism σ : F → (−, C′′) such that (−, g) = σ ∘ τ is a R-reflection of F. This proves (a)⇒(b). The converse is proven along the same lines, and (b)⇔(c) follows from Proposition 3.4, since F = R⃗. □

Remark 4.1. (1) When C = mod_{R} or C = FMod_{R} (the class of all finitely generated modules) then C has cokernels and therefore F is reflective in (C^{op}, Ab) .

(2) When C = proj_{R} (the class of all finitely generated projective modules) then (C^{op}, Ab) and Mod_{R} are equivalent, and thus in Proposition 4.1.(v) we can add the condition that proj_{R} has cokernels ([4, Proposition 3]).

(3) Straightforward variations on the above arguments prove that R is covariantly finite in fp (C^{op}, Ab) if and only if C has pseudo-cokernels (see [9]), and that the subcategory of representable covariant functors is reflective (resp. covariantly finite) in (C, Ab) if and only if C has kernels (resp. pseudo-kernels).

In the rest of this section, we let A be a complete abelian category; for a fixed object M ∈ A, we say that an object A ∈ A is M-cogenerated if there exists a monomorphism A → M^{I} for some set I, and we say that A is M-copresented if there exists an exact sequence 0 → A → M^{I} → M^{J} for some sets I, J. We denote by Cogen (M) and Copres (M) the classes of M-cogenerated and M-copresented objects in A, respectively, and by Prod (M) the class of all modules isomorphic to a product of copies of M.

Copres (M) can be thought of as the closure of M with respect to products and kernels of these products. So, in view of Corollary 3.2, Copres (M) is always contained in (and should be very close to) the smallest reflective subcategory of A containing M. It seems thus interesting to give necessary and sufficient conditions for Copres (M) to be a reflective subcategory of A. One such condition can be derived from [31, (16.4.7)] (note that Copres (M) is always closed for products), but we give it as a consequence of a more general result which may have independent interest.

Proposition 4.3. Let A be a complete abelian category, and let M be any object of A. Then Copres (M) is covariantly finite in A. In particular, when A is well-powered, Copres (M) is a reflective subcategory of A if and only if it is closed for kernels.

Proof. We prove that every object A of A has a Copres (M)-preenvelope, and the second part of the statement will then be a consequence of Corollary 3.2.

Let H = Hom (A, M) and let σ : A → M^{H} be the morphism induced by the family {h}_{h∈H}, that is h = π_{h} ∘ σ for every h ∈ H, where π_{h} : M^{H} → M are the canonical projections. Then it is easy to see that σ is a Prod (M)-preenvelope of A.

We claim that, if α : A → M^{I} is a morphism in A and (k_{α}, K_{α}) is the intersection of the kernels of all morphisms f : M^{I} → M such that f ∘ α = 0, then α factors through k_{α} and K_{α} ∈ Copres (M). The first assertion is clear and, if J = {f : M^{I} → M | f ∘ α = 0} and φ : M^{I} → M^{J} is such that π_{f} ∘ φ = f for every f ∈ J, where π_{f} : M^{J} → M is the canonical projection, then (k_{α}, K_{α}) is the kernel of φ and therefore K_{α} ∈ Copres (M).

In particular, σ factors as A →^{λ_{σ}} K_{σ} →^{k_{σ}} M^{H} with K_{σ} ∈ Copres (M), and we shall show that λ_{σ} is a Copres (M)-preenvelope. To see this, let β : A → F be any morphism in A with F ∈ Copres (M) and take an exact sequence

0 → F ⟶^{v} M^{I} ⟶^{g} M^{X}

in A. Now set α = v ∘ β : A → M^{I} and consider the factorization A →^{λ_{α}} K_{α} →^{k_{α}} M^{I} of α given by the above claim. If q_{x} : M^{X} → M are the canonical projections, then we get

(v, F) = ∩ {Ker (q_{x} ∘ g) | x ∈ X}.

Now, since q_{x} ∘ g ∘ α = q_{x} ∘ g ∘ v ∘ β = 0 for each x ∈ X, we see that k_{α} factors through v and therefore β factors through λ_{α}. So, in order to see that β factors through λ_{σ} it is enough to see that λ_{α} does. To prove this, let J and φ : M^{I} → M^{J} be as in the proof of the above claim and consider the diagram

A ⟶^{λ_{σ}} K_{σ} ⟶^{k_{σ}} M^{H}
∥ ↓^{g′} ↓^{g}
A ⟶^{λ_{α}} K_{α} ⟶^{k_{α}} M^{I} ⟶^{φ} M^{J}

Since σ is a Prod (M)-envelope, there exists g : M^{H} → M^{I} such that α = g ∘ σ. But now the morphism φ ∘ g : M^{H} → M verifies φ ∘ g ∘ σ = φ ∘ α = 0, and therefore φ ∘ g ∘ k_{σ} = 0 by the definition of k_{σ}; this implies that g ∘ k_{σ} factors through k_{α}, i.e. there exists g′ : K_{σ} → K_{α} with g ∘ k_{σ} = k_{α} ∘ g′, and then it is clear that λ_{α} = g′ ∘ λ_{σ}, which is the factorization we were seeking for. □

Next we list several sufficient conditions for Copres (M) to be reflective in A.

Proposition 4.4. Let A be a well-powered complete abelian category with enough injectives, and consider the following conditions on an object M of A:

(a) M has injective dimension at most 1 and Ext^{1} (M^{I}, M) = 0 for every set I.

(b) Ext^{1} (C, M) = 0 for each C ∈ Cogen (M).

(c) Hom (−, M) takes exact sequences of the form 0 → A → M^{I} → C → 0 with C ∈ Cogen (M) to exact sequences in Ab.

(d) Cogen (M) is closed for extensions.

(e) Copres (M) is a reflective subcategory of A.

Then (a)⇒(b)⇒(c)⇒(e) and (b)⇒(d)⇒(e).

Proof. (a)⇒(b). Let C ∈ Cogen (M) and take an exact sequence 0 → C → M^{I} → B → 0; now we have Ext^{1} (M^{I}, M) = 0 = Ext^{2} (B, M), and so Ext^{1} (C, M) = 0.

(b)⇒(c). This is clear.

(c)⇒(e). It is clearly enough to show that, if B ∈ Copres (M), then the kernel A of any morphism f : B → M^{I} is also in Copres (M). Since

A = ∩ {Ker h | h ∈ Hom (B, M) with A ⊆ Ker h},

by taking a copresentation 0 → B →^{v} M^{J} → M^{K}, the hypothesis implies that

A = ∩ {Ker (g ∘ v) | g ∈ Hom (M^{J}, M) with A ⊆ Ker (g ∘ v)}.

Taking now the composition A ↪ B →^{v} M^{J} and L = {g ∈ Hom (M^{J}, M) | A ⊆ Ker (g ∘ v)}, we get an exact sequence 0 → A → M^{J} ⟶ M^{L} which proves the claim.

(b)⇒(d). Let 0 → A → B → C → 0 be exact in A with A, C ∈ Cogen (M); taking the pushout of A → B and a monomorphism A → M^{I}, we get a commutative diagram with exact rows and vertical monomorphisms:

0 → A ⟶ B ⟶ C → 0
↓ ↓ ∥
0 → M^{I} ⟶ B̄ ⟶ C → 0

Now the lower row splits, so that B̄, and hence B, are in Cogen (M).

(d)⇒(e). It is enough to show that, if 0 → A → B → C → 0 is exact in A with B ∈ Copres (M) and C ∈ Cogen (M), then A ∈ Copres (M). For, we have an exact sequence 0 → B → M^{I} → D → 0 with D ∈ Cogen (M), and applying the cross lemma to these sequences we get new exact sequences 0 → C → E → D → 0 (and so, by (d), E ∈ Cogen (M)) and 0 → A → M^{I} → E → 0, which shows that A ∈ Copres (M). □

Example 4.1. (1) None of the non-stated implications in the above result is true. For instance, if M is a cogenerator of A then (d) and (e) trivially hold, while (b) and (c) hold only if M is injective. On the other hand, if S is a simple module which is finite dimensional over its endomorphism ring, then Cogen (S) consists of the semisimple Shomogeneous modules (cf. [23, Proposition 4.5]), and thus (c) and (e) hold, while (b) and (d) hold only if Ext^{1} (S, S) = 0, which is not always the case (e.g. take S = ℤ_{2} as a ℤ-module). Finally, if S is as above and Ext^{1} (S, S) = 0 then (b) holds, but S needs not have injective dimension ≤ 1.

(2) If M is an injective object of A then condition (a) in Proposition 4.4 holds, and hence Copres (M) is reflective in A. In that case, T = {T ∈ A | Hom (T, M) = 0} is a localizing subcategory of A and Copres (M) is just the subcategory of T -closed objects in the terminology of Gabriel [14]. Moreover, the left adjoint a : A → Copres (M) of the inclusion is exact and identifies Copres (M) with the quotient category A/T .

(3) A typical class of objects satisfying condition (a) in Proposition 4.4 is that of partially cotilting objects, i.e. objects M such that Cogen (M) ⊆ Ker Ext^{1} (−, M) and Ker Ext^{1} (−, M) is a hereditary torsion class (cf. the obvious generalization of [7, Lemma 2.6(b)]).

(4) If M_{R} is a quasi-injective module which is finitely generated over its endomorphism ring S then, by [6, Lemma 1.5] , M^{I} is M-generated for every set I. But then condition (c) in Proposition 4.4 holds since the exact sequences under consideration are in the category σ [M] of M-subgenerated modules (see [35, (16.1)]). Therefore Copres (M) is a reflective subcategory of Mod_{R}. If _{S}M is not finitely generated and we denote by τ_{M} (N) the trace of M in N, the M-copresented objects in σ [M] are those X fitting in an exact sequence of the form 0 → X → τ_{M} (M^{I}) → τ_{M} (M^{J}) for some sets I, J. They form a reflective subcategory of σ [M].

(5) In general, not even for A = Mod_{R} it is true that Copres (M) is a reflective subcategory of A, for a given object M. To show this we give the following proposition:

Proposition 4.5. The following assertions are equivalent for an artinian commutative ring R:

(a) R is quasi-Frobenius (i.e. self-injective).

(b) Copres (R) is a reflective subcategory of Mod_{R}.

(c) Copres (R) is closed for direct summands.

Proof. The implications (a)⇒(b)⇒(c) are clear. To see that (c)⇒(a), it is not restrictive to assume that R is also local. Suppose R is not quasi-Frobenius. Then the simple module S is not reflexive [1, (30.8)] and thus, by [21, Exercise (12), p. 331], we have Soc (S) ≅ S^{n} for some integer n ≥ 2 and S^{∗} = Hom_{R} (S, R) ≅ S^{n}. Now, by using the fact that R^{I} is projective for each set I, one easily proves that any non-zero finitely generated module of Copres (R) must be isomorphic to X^{∗} for some non-zero finitely generated R-module X. On the other hand, such an X^{∗} contains a copy of S^{∗}, and thus it cannot be isomorphic to S. Hence S^{n} ≅ S^{∗} ∈ Copres (R) and S ∉ Copres (R). □

Remark 4.2. All results about Copres (M) admit straightforward dualizations to a cocomplete abelian category A. Hence, the subcategory Pres (M) of M-presented objects is coreflective in A if and only if it is closed for cokernels. When A = Mod_{R} that includes, as particular situations, the cases when M_{R} is tilting [17], M_{R} is Σ-quasiprojective [16] or, more generally, M_{R} is w-Σ-quasiprojective in the sense of [6].

5. Reflectivity of Add (M)

An additive category C will be called a generalized Krull-Schmidt category when every object in C is a direct sum of indecomposable objects with local endomorphism ring and, moreover, the isomorphism classes of indecomposable objects form a set. In this section, we study reflective generalized Krull-Schmidt subcategories of Mod_{R}. It is clear that it is equivalent to study reflective subcategories of the form Add (M_{R}) (i.e., the subcategory of direct summands of coproducts of copies of a module M_{R}).

By [23], Add (M) is closed for products if and only if it is covariantly finite in Mod_{R}, in which case M is called product-complete. Such a module is always Σ-pure-injective, so that Add (M) consists in that case of the direct summands of products of copies of M_{R}. In particular, Add (M) ⊆ Copres (M), and the converse inclusion holds exactly when Add (M) is reflective in Mod_{R}:

Proposition 5.1. Let M_{R} be any module. Then the following assertions are equivalent:

(a) Add (M) is reflective in Mod_{R}.

(b) Add (M) is closed for products and kernels.

(c) Add (M) = Copres (M).

Proof. (a)⇔(b). Follows from Theorem 3.1 and [23, Theorem 3.1].

(b)⇒(c). Condition (b) implies Copres (M) ⊆ Add (M), and hence equality.

(c)⇒(a). We have to show that Add (M) = Copres (M) is closed for kernels, and it is enough to see that, if 0 → A → B → C → 0 is exact with B ∈ Copres (M) and C ∈ Cogen (M), then A ∈ Copres (M). But we have a split sequence 0 → B → M^{I} → D → 0 with D ∈ Cogen (M), and by applying the cross lemma we get a split sequence 0 → C → X → D → 0, whence X ∈ Cogen (M), and an exact sequence 0 → A → M^{I} → X → 0, which shows that A ∈ Copres (M). □

For the rest of this section, we fix a nonzero module M_{R} and set S = End (M_{R}) and R̄ = BiEnd (M_{R}) = End (_{S}M). We write ρ : R → R̄ for the canonical ring homomorphism, and denote by ρ_{∗} : ModR̄ → Mod_{R} the restriction of scalars functor.

In the examples that will come, we will frequently consider the case when M = R⊕X for

certain module X_{R}. Then there is a canonical ring isomorphism S ≅ ( \begin{matrix}R & Hom_{R} (X, R) \\ X & End (X_{R}) \end{matrix} )

and, setting e = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ S, we have _{S}M ≅ Se and thus R̄ ≅ eSe ≅ R.

The proof of the following lemma is left to the reader.

Lemma 5.2. The functor ρ_{∗} induces an equivalence of categories Add (MR̄) ≅ Add (M_{R}).

Lemma 5.3. M_{R} is product complete if and only if MR̄ is product complete.

Proof. The ”if” part is trivial. For the converse, fix a set I; by the preceding lemma there is a module in Add (MR̄) whose restriction of scalars is isomorphic to M_{R}^{I}. So we have a right R̄-module structure on the abelian group M^{I} such that M^{I} ∈ Add (MR̄). Now, by Lemma 5.2, the canonical projections p_{i} : M^{I} → M are R̄-homomorphisms and hence, for any m = (m_{i})_{i∈I} ∈ M^{I} and α ∈ R̄, we have p_{j} (mα) = (p_{j}m) α = m_{j}α and therefore (m_{i})_{i∈I} α = (m_{i}α)_{i∈I}. This means that the described structure of M^{I} is the canonical one (induced by the natural structure on MR̄), and thus MR̄ is product-complete. □

Theorem 5.4. Add (M_{R}) is reflective in Mod_{R} if and only if Add (MR̄) is reflective in ModR̄. When these conditions hold, we have:

(i) _{S}M is projective and finitely generated.

(ii) R̄ is right Σ-pure-injective.

(iii) The canonical map ρ : R_{R} → R̄_{R} is the Add (M_{R})-reflection of R_{R}.

Proof. If Add (MR̄) is reflective in ModR̄ then M_{R} is product-complete by Lemma 5.3, and we only have to check that every R-homomorphism f : X → Y with X, Y ∈ Add (M_{R}) has its kernel in Add (M). But Lemma 5.2 tells us that f is also a R̄-homomorphism. Hence Ker f ∈ Add (MR̄) and, by restricting scalars, Ker f ∈ Add (M_{R}). For the converse, reverse the argument.

Suppose now that Add (M_{R}) is reflective in Mod_{R}, and let us prove:

(i) Let λ : R → X is an Add (M)-reflection; since R_{R} is finitely generated, it is easy to see that X is a direct summand of M^{n} for some positive integer n. Now, the image of λ by the functor Hom_{R} (−, M) is an isomorphism Hom_{R} (X, M) ≅ M and it is clear that Hom_{R} (X, M) is projective and finitely generated as a left S-module.

(ii) Since MR̄ is product-complete, Add (MR̄) consists of Σ-pure-injective modules, and since MR̄ is a generator [34, (IV.6.7)] we have R̄R̄ ∈ Add (MR̄).

(iii) Lemma 5.2 implies that R̄_{R} ∈ Add (M_{R}), and it is easy to see that the composition of the natural maps

MR̄ ⟶^{≅} HomR̄ (R̄, M) = Hom_{R} (R̄, M) ⟶^{Hom_{R}(ρ,M)} Hom_{R} (R, M) ⟶^{≅} M_{R}

is the identity. Therefore Hom_{R} (ρ, M) is an isomorphism and it follows that ρ is an Add (M_{R})-reflection. □

Remark 5.1. If Add (M_{R}) is reflective in Mod_{R} then R → R̄ is the ring of definable scalars corresponding to the definable subcategory Add (M_{R}) (see [22, Corollary 11.10]).

Corollary 5.5. The following assertions are equivalent:

(a) M_{R} is indecomposable and Add (M_{R}) is reflective in Mod_{R}.

(b) S is a division ring and _{S}M is finite dimensional.

Proof. (a)⇒(b). From [23, Theorem 4.1], [8, Proposition 4.4] and our Theorem 5.4, one gets that _{S}M is free of finite rank and S is left artinian and local. Hence _{S}MR̄ defines a Morita equivalence and Add (MR̄) = ProjR̄. Therefore rgD (S) = rgD \binom{}{R̄} ≤ 2, which implies that S is a division ring.

(b)⇒(a). Since R̄ is semisimple, the implication is clear. □

A module M_{R} is called self-small when, for every set I, the canonical homomorphism Hom_{R} (M, M)^{(I)} → Hom_{R} (M, M^{(I)}) is an isomorphism. It is clear that M_{R} is self-small if and only if so is MR̄. We shall say that a ring A is right pro-reflective when Proj_{A} is reflective in Mod_{A}, i.e., when A is right perfect, left coherent and rgD (A) ≤ 2.

Proposition 5.6. Let M_{R} be self-small. Then the following assertions are equivalent: (a) Add (M_{R}) is reflective in Mod_{R}.

(b) S is right pro-reflective and _{S}M is finitely generated projective.

In case these conditions hold, we have:

(i) R̄ is right perfect.

(ii) rgD \binom{}{R̄} ≤ 2 + pd (MR̄).

(iii) Assume that MR̄ is finitely generated. Then R̄ is left coherent if and only if Hom_{R} (M, R̄) is a finitely generated left R̄-module.

Proof. The equivalence of (a) and (b) is proven as Proposition 3.8 in [23], using Theorem 5.4.(i). For the rest, by replacing M by some power M^{n} if necessary, we can assume that M = R̄ ⊕ X for certain X ∈ Mod_{R}. Now we can prove:

(i) Since S is right perfect and S = End (R̄ ⊕ X) ≅ ( \begin{matrix}R̄ & Hom (X, R̄) \\ X & End (X) \end{matrix} ), we get that

R̄ is right perfect [1, (28.6)].

(ii) For each NR̄, we get an exact sequence 0 → K → P_{1} → P_{0} → N → 0 in ModR̄ with P_{0} and P_{1} projective. Since MR̄ is a generator, we have ProjR̄ ⊆ Add (MR̄), and therefore K ∈ Add (MR̄), which clearly implies that pd (KR̄) ≤ pd (MR̄), whence the result follows.

(iii) Assume that MR̄ is finitely generated and (a),(b) hold, so that R̄ is right perfect. If R̄ is left coherent then it is indeed strongly left coherent (or left Π-coherent) in the sense of [13], and therefore Hom (M, R̄) is a finitely generated left R̄-module by [5, Theorem 1].

Conversely, suppose that Hom (M, R̄) is a finitely generated left R̄-module. Looking

S as matrices in the above sense, we consider the element e = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ∈ S and the

bimodule R̄ (eS)_{S}. In R̄ Mod we have eS ≅ R̄ ⊕ Hom_{R} (X, R̄), and since this is finitely generated by the hypothesis, [6, Lemma 1.5] says that (eS)^{I} is eS-generated in Mod_{S}. Now, since the functor Hom_{S} (eS, −) : Mod_{S} → ModR̄ takes projective eS-generated right S-modules to projective right R̄-modules, we infer that R̄^{I} ≅ Hom_{S} (eS, (eS)^{I}) is

projective in ModR̄, so that R̄ is left coherent. □

Example 5.1. (1) In the situation of Proposition 5.6 we can have rgD \binom{}{R̄} = ∞ and R̄ not left coherent. Indeed, let us consider a non-Artinian commutative local ring R with

J^{2} = 0, where J is the radical (e.g. R = \frac{K [x_{1}, ...x_{n}, ...]}{(x_{1}, ...x_{n}, ...)^{2}} with K a field). We propose

the reader to prove that, if {e_{1}, ..., e_{t}} is a finite family of elements in R^{I} (for some set I) which is R/J-linearly independent modulo J^{I}, then it is R-linearly independent and thus ∑_{i=1}^{t} e_{i}R = ⊕_{i=1}^{t}e_{i}R is a projective submodule of R^{I}. Once this is done, if {e_{α} | α ∈ A} is a maximal family of R/J-linearly independent elements of R^{I} modulo J^{I} and {f_{β} | β ∈ B} is a maximal family of R/J-linearly independent elements of J^{I} modulo R^{I}J, then one easily sees that R^{I} = (⊕e_{α}R) ⊕ (⊕f_{β}R), so that R^{I} is a direct sum of a free and a semisimple submodule. If now M = R ⊕R/J, then Add (M) is clearly reflective in Mod_{R}, but R̄ ≅ R has infinite global dimension and is not coherent.

(2) We do not know if a module M_{R} satisfying either (a) or (b) in Proposition 5.6 is necessarily self-small, but it needs not be finitely generated over R̄. Indeed, if R is the Kronecker algebra over a field K, then the generic module G_{R} (cf. [20]) satisfies End (G_{R}) ≅ K (X) and dim_{K(X)} G = 2. So G_{BiEnd(G)}, and consequently G_{R}, is self small. Now set M = R ⊕ G, so that R̄ = R and M_{R} is not finitely generated; since Hom (G, R) = 0, it follows easily that (a-b) in Proposition 5.6 hold for M.

(3) In Proposition 5.6, it is easy to give examples where both S and R̄ are right proreflective but M_{R} is not an endogenerator. For instance, take any right pro-reflective ring R and a module X_{R} such that D = End (X_{R}) is a division ring and _{D}X is finitedimensional (if R is a C-algebra such that R/J (R) is an Artin C-algebra, then every simple right R-module has that property). If X_{R} is not a direct summand of any kernel of a morphism between projectives, then M = R ⊕ X satisfies all the above conditions.

In what follows, we shall deal with the special case in Theorem 5.4 when _{S}M is a (pro)generator. For this purpose we define:

Definition 5.1. A module M_{R} will be called a reflective endogenerator when Add (M_{R}) is reflective in Mod_{R} and _{S}M is a generator. Two reflective endogenerators M_{R} and N_{R} will be called equivalent when Add (M_{R}) = Add (N_{R}).

In this situation MR̄ is also a progenerator, so that Add (MR̄) = ProjR̄ and S and R̄ are Morita equivalent rings. By Proposition 5.6, both rings are right pro-reflective. On the other hand, since direct summands of M_{R} and MR̄ are the same, we have M = M_{1}^{n_{1}} ⊕ ⋯ ⊕ M_{t}^{n_{t}}, where the n_{i}’s are positive integers and M_{1}, ..., M_{t} are pairwise nonisomorphic indecomposable modules.

Definition 5.2. A ring homomorphism φ : R → A will be called a right pseudoepimorphism when the canonical ring homomorphism A ≅ End (A_{A}) → End (A_{R}) induced by φ_{∗} is onto (and hence, an isomorphism). Two right pseudoepimorphisms φ : R → A and φ′ : R → A′ will be said to be equivalent when there is a ring isomorphism ϕ : A → A′ such that ϕ ∘ φ = φ′.

If φ : R → A is a right pseudoepimorphism then A = BiEnd (A_{R}) and φ is (equivalent to) the canonical map R → BiEnd (A_{R}); by Proposition 5.6, A_{R} is a reflective endogenerator if and only if A is right pro-reflective. In fact, we have:

Proposition 5.7. Let R be any ring. There is a bijective correspondence between:

(a) the set of equivalence classes of reflective endogenerators in Mod_{R}; and

(b) the set of equivalence classes of right pseudoepimorphisms φ : R → A with A right pro-reflective.

Proof. If Add (M_{R}) is reflective then R̄R̄ ∈ Add (MR̄) by Theorem 5.4, and End \binom{}{R̄R̄} = End \binom{}{R̄_{R}} by Lemma 5.2, so that the ring homomorphism ρ : R → R̄ is a right pseudoepimorphism. Note that, if Add (M_{R}) = Add (N_{R}), then BiEnd (M_{R}) = BiEnd (N_{R}) [1, (14.1)], and thus ρ does not depend on M_{R} but on its equivalence class. If, moreover, M_{R} is a reflective endogenerator, then R̄ is right pro-reflective ant thus the assignation M_{R} ↦ ρ establishes a map from (a) to (b).

Conversely, if φ : R → A is as in (b), then Add (A_{R}) is reflective in Mod_{R} by the remark above, so that A_{R} is a reflective endogenerator in Mod_{R}. Clearly, the assignation φ ↦ A_{R} depends only on the equivalence class of φ, and so it establishes a map from (b) to (a) which is an inverse for M_{R} ↦ ρ. □

Remark 5.2. (1) If φ is a right pseudoepimorphism then the restriction of φ_{∗} to Add (A_{A}) is a full functor. In particular, when the ring A is semisimple, every pseudoepimorphism R → A is an epimorphism (cf. [34, (XI.1.2)]). Thus, in view of Corollary 5.5, the above proposition extends Proposition 2.1 in [27].

(2) The bijection of Proposition 5.7 induces another one between the set of: (a) equivalence classes of flat reflective endogenerators; and (b) isomorphism classes of flat ring epimorphisms φ : R → A with A right pro-reflective. Indeed, if M_{R} is a flat reflective endogenerator then every module in Add (M_{R}) (and in particular R̄_{R}) is flat, and thus all we need to prove is that a flat right pseudoepimorphisms φ : R → A with A right pro-reflective is an epimorphism. But, in that case, A ⊗_{R} A is projective (and hence torsionless) in Mod_{A} and the multiplication map μ : A ⊗_{R} A → A is carried to an isomorphism by the functor Hom_{A} (−, A), so that it is an isomorphism and therefore φ is a ring epimorphism by [34, (XI.1.2)].

(3) When the ring R is commutative, there is a bijective correspondence between the sets of: (a) equivalence classes of reflective endogenerators in Mod_{R}; (b) equivalence classes of ring epimorphisms φ : R → A where A is a finite direct product of fields; and (c) finite parts of Spec (R). Indeed, one proves along the lines of [32, Corollary 1.2] that if R → A is a right pseudoepimorphism then the commutativity of R implies that of A, so that pro-reflectivity of A means that it is a finite direct product of fields. The rest follows the pattern of [27, Example 1.2.a]. The bijection (b)↔(c) takes φ : R → K_{i} × ⋯ × K_{n} to {Ker (π_{i} ∘ φ) | i = 1, ..., n} (where π_{i} : ∏ K_{i} → K_{i} is the canonical projection) in one direction and {P_{1}, ..., P_{n}} to R → Q (R/P_{1})×⋯×Q (R/P_{n}) (where Q (D) is the quotient field of the domain D) in the other.

Example 5.2. (1) Let D be a commutative integral domain with quotient field Q; then

the inclusion R = ( \begin{matrix}D & D \\ 0 & D \end{matrix} ) ↪ ( \begin{matrix}Q & Q \\ 0 & Q \end{matrix} ) = T is a ring epimorphism and, since T is

left and right hereditary and artinian, T_{R} and _{R}T are both reflective endogenerators.

(2) Let R be a right artinian ring and let M_{R} be such that Hom_{R} (−, M) is an object of finite length in the category (mod_{R}^{op}, Ab). Let I_{M} be a family of representatives of the isomorphism classes of indecomposable modules X ∈ mod_{R} with Hom_{R} (X, M) ≠ 0; by [3, Theorem 2.12] we know that I_{M} is finite, and we set M̃ = ⊕_{X∈I_{M}} X.

We claim that Add \binom{}{˜M} is reflective in Mod_{R}, but M̃ is an endogenerator if and

only if M̃_{R} is projective. To prove that we first replace M_{R} by M̃_{R} if necessary and assume from the beginning that, for an indecomposable X ∈ mod_{R}, Hom_{R} (X, M) ≠ 0 implies X ∈ Add (M_{R}). In particular, if ϵ : P → M is the projective cover of M_{R} then P ∈ Add (M_{R}) and so the class of indecomposable projective direct summands of M_{R} is not empty. We pick a set {P_{1} = e_{1}R, ..., P_{t} = e_{t}R} of representatives of the isomorphism classes of these direct summands, where {e_{1}, ..., e_{t}} are pairwise orthogonal idempotents of R. One immediately gets that, if S = End (M_{R}), then _{S}M = Se_{1} ⊕ ⋯ ⊕ Se_{t} (viewing the e_{i} as elements of M_{R}, since e_{i}R is a direct summand of M_{R} for each i = 1, ..., t). Also, clearly, Se_{i} ≅ Sϵ_{i}, where ϵ_{i} is the idempotent endomorphism M_{R} → e_{i}R → M_{R}. Hence SM is finitely generated projective. On the other side, again by [3, Theorem 2.12], S is left artinian and so M_{R} has finite endolength, which implies that Add (M_{R}) is closed for products and direct limits (cf. [23, Theorem 4.1]).

If now f : M^{(I)} → M^{(J)} is a R-homomorphism, then it is the direct limit of a system of morphisms f_{α} : M^{n_{α}} → M^{(J)}, whose kernels are in Add (M_{R}) by our choice of M_{R}. Hence Ker f = \underrightarrow{lim} Ker f_{α} is in Add (M_{R}) and therefore Add (M_{R}) is reflective in Mod_{R}. Moreover, _{S}M is a generator if and only if all indecomposable direct summands of _{S}S appear as direct summands of _{S}M. This is equivalent to say that all indecomposable direct summands of M_{R} belong, up to isomorphism, to the family {P_{1}, ..., P_{t}}, i.e., that M_{R} is projective.

In view of the previous example, a natural question arises: Suppose that Add (M_{R}) is reflective in Mod_{R}; are there only finitely many non-isomorphic indecomposable direct summands of M_{R}? Note that if R is right pure-semisimple and M_{R} is the direct sum of all indecomposable right R-modules, an affirmative answer would solve the “pure-semisimple conjecture” (see [19] and [33] for the most recent advances) Hence, the question seems difficult and we can only give the following partial affirmative answers:

Proposition 5.8. Let R be a principal ideal domain, Q its quotient field and M an arbitrary R-module. Then Add (M_{R}) is reflective in Mod_{R} if and only if there are finitely many (and perhaps none) pairwise nonassociated prime elements p_{1}, ..., p_{t} in R, integers n_{1}, ..., n_{t} ≥ 1 and sets I_{11}, ..., I_{1n_{1}}, ..., I_{t1}, ..., I_{tn_{t}}, J (non-empty except perhaps J) such that

M ≅ (\frac{R}{(p_{1})})^{(I_{11})} ⊕ ⋯ ⊕ (\frac{R}{(p_{1}^{n_{1}})})^{(I_{1n_{1}})} ⊕ ⋯ ⊕ (\frac{R}{(p_{t})})^{(I_{t1})} ⊕ ⋯ ⊕ (\frac{R}{(p_{t}^{n_{t}})})^{(I_{tn_{t}})} ⊕ Q^{(J)}.

Proof. Assume first that M has the given form; then by [23, Theorem 4.1] each summand is product-complete and hence so is M; since Add (M) is clearly closed for kernels, it is reflective in Mod_{R}.

For the converse we can assume that M is of the form M = ⊕M_{α} for some set {M_{α} | α ∈ A} of pairwise non-isomorphic indecomposable Σ-pure-injective R-modules. Now, by [20, (8.56)], the indecomposable Σ-pure-injective R-modules are: (a) the quotient field Q; and, for every prime element p ∈ R, (b_{1}) the modules R/ (p^{n}) with n ≥ 1; and (b_{2}) the Prüfer module R/ (p^{∞}) (the p-adic module R̂_{p} is not Σ-pure-injective since it is not an artinian ring). Note that, for a fixed prime p, Add (M) cannot contain a copy of R/ (p^{n}) for each n ≥ 1, since Add (M) is closed for inverse limits and it does not contain R̂_{p}. On the other hand, the fact that Add (M) is closed for kernels implies that, if R/ (p^{n}) is among the M_{α}’s, then so is R/ (p^{i}) for each i = 1, 2, ..., n. And it also implies that no R/ (p^{∞}) is a direct summand of M, because otherwise Add (M) would contain every R/ (p^{n}). So, we only have to prove that the number of distinct R/ (p) that can appear among the M_{α}’s is finite. But if R/ (p_{i}) ∈ Add (M) for infinitely many primes {p_{1}, p_{2}, ...} then we would have ⊕_{i=1}^{∞}R/ (p_{i}) ∈ Add (M), so that the pure monomorphism ⊕_{i=1}^{∞}R/ (p_{i}) → ∏_{i=1}^{∞} R/ (p_{i}) would split. □

Proposition 5.9. Let R be a finite dimensional tame hereditary algebra over a field K which is not representation-finite, and let G be the generic module. If Add (M_{R}) is reflective in Mod_{R}, then there are finitely generated indecomposable R-modules X_{1}, ..., X_{t} and (possibly empty) sets I_{1}, ..., I_{t} and J such that

M_{R} ≅ X_{1}^{(I_{1})} ⊕ ⋯ ⊕ X_{t}^{(I_{t})} ⊕ G^{(J)}.

Proof. Following the terminology of [29], this is proved as Proposition 5.8 by taking S, S [n], S [∞], Ŝ and G instead of R/ (p), R/ (p^{n}), R/ (p^{∞}), R̂_{p} and Q, where S is a simple regular module and G is the generic module. □

Remark 5.3. Notice the difference between reflectivity of Add (M) in Mod_{R} and reflectivity of add (M) in mod_{R}. If R is a right artinian ring and M_{R} is the direct sum of one copy of each indecomposable R-module X_{R} such that Hom_{R} (−, X) is an object of finite length in (mod_{R}^{op}, Ab) (cf. Example 5.2), then add (M_{R}) is reflective in mod_{R} (and hence \overrightarrow{add (M_{R})} is reflective in Mod_{R} by Proposition 3.4). However, it is well known (cf. [28]) that, when R is a hereditary finite dimensional algebra over a field that is not representation-finite, M_{R} has infinitely many non-isomorphic direct summands. Then Proposition 5.9 says that Add (M_{R}) is not reflective in Mod_{R} when R is tame and not representation-finite.

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(Juan Rada) Departamento de Matemáticas, Universidad de los Andes, 5101 Mérida, Venezuela. E-mail address: juanrada@ciens.ula.ve

(Manuel Saorín and Alberto del Valle) Departamento de Matemáticas, Universidad de Murcia, Aptdo. 4021, Espinardo 30.100, Murcia, Spain.

E-mail address: msaorinc@fcu.um.es, alberto@fcu.um.es