ON MINIMAL APPROXIMATIONS OF MODULES

HENNING KRAUSE AND MANUEL SAORÍN

1. Left minimal morphisms

Let R be a an associative ring with identity. We consider the category Mod R of (right) R-modules and denote by mod R the full subcategory of finitely presented R-modules. A morphism ϕ: M → N in any category is said to be left minimal if any endomorphism ψ of N satisfying ψ^{∘} ϕ = ϕ is an isomorphism. A concept which is closely related is that of an injective envelope in abelian category. Recall that a monomorphism ϕ: M → N in any abelian category is an injective envelope of M provided that N is an injective object and every morphism N → N′ is a monomorphism if the composition with ϕ is a monomorphism.

Lemma 1.1. A monomorphism ϕ: M → N into an injective object N is an injective envelope if and only if ϕ is left minimal.

Proof. Straightforward.

The following result gives the precise relation between left minimal morphisms and injective envelopes.

Proposition 1.2. Let ϕ: M → N be a morphism in some additive category A and suppose that idempotents in End(N) split. Then the following are equivalent:

(1) There exists a decomposition ϕ = (ϕ′, ϕ′′): M → N′ ∐ N′′ = N such that ϕ′ is left minimal and ϕ′′ = 0.

(2) There exists an abelian category B and an additive functor F: A → B such that

(a) F (N) is injective;

(b) F induces an isomorphism Hom_{A}(M ∐ N, N) ⟶ Hom_{B}(F (M ∐ N), F (N);

(c) the image Im F (ϕ) has an injective envelope.

A decomposition ϕ = (ϕ′, ϕ′′): M → N′ ∐ N′′ = N of a morphism ϕ such that ϕ′ is left minimal and ϕ′′ = 0 is unique up to isomorphism. The morphism ϕ′ is sometimes called the minimal version of ϕ.

Proof of Proposition 1.2. (1) ⇒ (2) Let S = End(M ∐ N)^{op} and consider the functor

F : A ⟶ B = (Mod − S)^{op}, X ⟶ Hom(X, M ∐ N)

It is clear that (a) and (b) are satisfied. A decomposition ϕ = (ϕ′, ϕ′′): M → N′ ∐ N′′ = N of ϕ such that ϕ′ is left minimal and ϕ′′ = 0 induces a morphism Im F (ϕ) → F (N′) in B which is an injective envelope by Lemma 1.1.

(2) ⇒ (1) Let F: A → B be as in the statement and suppose that ε: Im F (ϕ) → Q is a injective envelope. Consnider the induced morphism ψ: Im F (ϕ) → F (N). There are morphisms α: Q → F (N) and β: F (N) → Q satisfying α^{∘} ε = ψ and β^{∘} ψ = ε since Q and F (N) are injective. We may assume that β^{∘} α = id_{Q} since ε is an injective envelope. Thus α^{∘} β is an idempotent in End(F (N)) and we obtain therefore a decomposition N = N′ ∐ N′′ in A with F (N′) = Im α. It is now easily checked that in the induced decomposition ϕ = (ϕ′, ϕ′′): M → N′ ∐ N′′ = N of ϕ the morphism ϕ′ is left minimal and ϕ′′ = 0.

Our main application of the preceding result depends on the well-known fact that the fully faithful functor

Mod R ⟶ D(R) = (mod R^{op}, Ab), M ↦ M ⊗_{R} −

from Mod R into the category of additive functors mod R^{op} → Ab identifies the pure-injective R-modules with the injective objects in D(R).

Corollary 1.3. Let ϕ: M → N be a morphism in Mod R and suppose that N is pure-injective. Then there exists a decomposition

ϕ = (ϕ′, ϕ′′): M ⟶ N′ ∐ N′′ = N

such that ϕ′ is left minimal and ϕ′′ = 0.

Proof. The Grothendieck category D(R) has injective envelopes, and the assertion is therefore a consequence of Proposition 1.2.

Another application of Proposition 1.2 is the following.

Corollary 1.4. Let ϕ: M → N be a morphism in some additive category and suppose that idempotents in End(N) split. If N is a finite coproduct of indecomposable objects with local endomorphism ring, then there exists a decomposition

ϕ = (ϕ′, ϕ′′): M ⟶ N′ ∐ N′′ = N

such that ϕ′ is left minimal and ϕ′′ = 0.

Proof. We consider the functor

F: A ⟶ B = (Mod S)^{op}, X ↦ Hom(X, M ∐ N)

where S = End(M ∐ N)^{op}. The assumption on N implies that soc F (N) is essential in F(N) and it is a finite coproduct of simple objects, and we find a decomposition N = N′ ∐ N′′ such that soc Im F (ϕ) = soc F (N′). The induced map Im F (ϕ) → F (N′) is an injective envelope and the assertion is now a consequence of Proposition 1.2.

Remark 1.5. We leave it to the reader to formulate the dual statements of Proposition 1.2 and Corollary 1.4. For instance, the dual of Corollary 1.4 generalizes the well-known fact that any finitely generated module over a semi-perfect ring has a projective cover.

Another application of Proposition 1.2 has to do with morphisms ending at torsion-injectives. Let A be an abelian category and T ⊂ Ob(A) a class of objects closed under taking subobjects and quotients. Suppose that, as a category, T has enough injectives. When viewed as objects of A, the injective objects of T will be called torsion-injective objectcs (relative to) T . We have now:

Corollary 1.6. With the above terminology, let ϕ: M → N be a morphism in A and suppose that N is torsion-injective relative to T . If the inclusion functor i : T ↪ A has a left adjoint r : A ⟶ T and the counit of the adjunction is an epimorphism, then the following assertions are equivalent:

(1) There exists a decomposition

ϕ = (ϕ′, ϕ′′): M ⟶ N′ ∐ N′′ = N

such that ϕ′ is left minimal and ϕ′′ = 0.

(2) Imϕ has an injective envelope in T

Proof. (1)⟹(2) Bearing in mind that r(N)≅N, it is straightforward.

(2)⟹(1) Apply proposition 1.2, with F=r Example 1.7. We list a few examples of a class of objects T in an abelian category, so that T has enough injectives.

(1) Let A be an abelian category with enough injectives. If T ⊆ Ob(A) is closed for subobjects an quotients and we assume that the inclusion functor i : T ↪ A has a right adjoint t : A ⟶ T , then T has enough injectives. Indeed, if T ∈ T and T ⟶^{λ} E is a monomorphism into an injective object of E, then t(λ) is also a monomorphism and t(E) is an injective object of T . Moreover, the adjunction equations yield a (split) monomorphism i(T) ⟶ i ∘ t ∘ i(T), whence a monomorphism T ⟶^{μ} t(T). Now, λ ∘ μ is the desired embedding of T into an injective object of T . Two particular instances of this sitution are:

(i) When A is a Grothendieck category and T is closed for subob-

jects, quotients and direct sums.

(ii) When R is an Artin algebra or, more generally, an (Artinian)

ring with selfduality (see [1, p. 332], for the definition), taking A = mod − R and T ⊆ mod − R any class closed for submodules, quotients and (finite) direct sums.

In case (i) Corollary 1.6 applies, for instance, when T is also closed for products. In (ii) Corollary 1.6 always applies.

(2) Let R be an Artin algebra and A = mod(mod − R) the abelian category of finitely presented contravariant functors mod−R ⟶ Ab. We take T = mod_{0}(mod− R) the full subcategory of those functors which vanish on projective modules. By [3, Theorem 7.5], we know that T has enough injectives (actually, even injective envelopes) and the torsion-injective objects relative to T are the Ext_{R}^{1}(−, M), with M ∈ mod − R. Since the inclusion functor has both a left and a right adjoint, Corollary 1.6 applies here. This has a homological consequence.

For R an Artin algebra, let us say that an extenxion ξ : 0 → M ⟶ Y ⟶ X → 0 is left minimal if, whenever α ∈ End_{R}(M) is an endomorphism such that Ext_{R}^{1}(X, α)(ξ) = ξ, we have that α is an isomorphism. We have now:

Corollary 1.8. Let R be an Artin algebra and ξ : 0 → M ⟶^{f} Y ⟶^{g} X → 0 an exact sequence in mod-R. There exists a decomposition f = f′ ∐ f′′ : M′ ∐ M′′ ⟶ Y′ ∐ Y′′ such that f” is an isomorphism and the induced extension ξ′ : 0 → M′ ⟶^{f′} Y′ ⟶^{g′} X → 0 is left minimal. In particular, ξ is left minimal if, and only if, f ∈ rad(M, Y).

Proof. View ξ as morphism Hom_{R}(−, X) ⟶ Ext_{R}^{1}(−, M) in the category mod(mod-R) and apply Corollary 1.6, bearing in mind that, by [3, Theorem 7.5], every object of T = mod_{0}(mod − R) has an injective envelope.

2. Left approximations

Let A be any category and C be any class of objects in A. Following [4], a morphism M → N in A is called a left C-approximation of M provided that N belongs to C and the induced morphism Hom(N, C) → Hom(M, C) is surjective for every C in C. If every object in A has a left C-approximation, then C is said to be covariantly finite in A. Note that this concept has been introduced independently in [7].

We begin our discussion with a criterion for a class of modules to be covariantly finite. Suppose that C is a class of R-modules which is closed under direct summands. It is easily seen that C needs to be closed under taking products if C is covariantly finite in Mod R. More generally, in [15, Corollary 3.5(c)] the following lemma is proved, leaning upon a result of Kielpinski [12] wrongly attributed to Enochs.

Lemma 2.1. Let C be any class of R-modules which is closed under pure submodules. Then the following are equivalent:

(1) Every R-module has a left C-approximation.

(2) C is closed under taking products.

The next result is essentially a reformulation of Corollary 1.3.

Theorem 2.2. Let C be any class of modules which is closed under direct summands. Let M be an R-module and suppose that M has a left C-approximation which is pure-injective. Then M has a minimal left C-approximation.

Proof. Apply Corollary 1.3.

We present an interesting consequence.

Corollary 2.3. Let C be any class of pure-injective R-modules which is closed under direct summands. Then the following are equivalent:

(1) Every R-module has a left C-approximation.

(2) Every R-module has a minimal left C-approximation.

(3) C is closed under taking products.

Proof. (1) ⇒ (2) Use Theorem 2.2.

(2) ⇒ (3) Use Lemma 2.1.

(3) ⇒ (1) Let D be the class of pure submodules of modules in C. Then every R-module M has a left D-approximation M → N by Lemma 2.1. Suppose that N is the pure submodule of N′ ∈ C. It is clear that the composition M → N′ is a left C-approximation.

Example 2.4. (1) Let C be the class of all pure-injective R-modules. Then a morphism ϕ: M → N is a minimal left C-approximation if and only if ϕ is a pureinjective envelope of M (i.e. N is pure-injective and any map ψ: N → N′ is a pure monomorphism if and only if ψ^{∘} ϕ is a pure monomorphism). In deed, the canonical map ϕ_{M}: M → D^{2}M (where D = Hom_{ℤ}(−, ℚ/ℤ)) is a left C-approximation since D^{2}M is pure-injective and ϕ_{M} is a pure monomorphism. Therefore a minimal left Capproximation M → N is a pure monomorphism, hence a pure-injective envelope. Conversely, if ϕ: M → N is a pure-injective envelope, then any endomorphism ψ of N satisfying ϕ = ψ^{∘} ϕ needs to be a pure-monomorphism, hence a split monomorphism, and therefore an isomorphism. The fact that every module has a pure-injective envelope is a classical result of Kiełpiński [12].

(2) Let C be the class of all injective R-modules. Then a morphism ϕ: M → N is a minimal left C-approximation if and only if ϕ is an injective envelope of M.

Concerning torsion-injective left approximations, we have:

Proposition 2.5. Let G be a Grothendieck category and T ⊆ Ob(G) a class of objects closed under subobjects, quotients and having enough injectives (e.g., if T is also closed for direct sums). The following assertions are equivalent:

(1) T is closed for direct products

(2) The class _{T} Inj of torsion-injective objects is closed for direct products in G (3) Every object of G has a (minimal) left T -approximation

(4) Every object of G has a (minimal) torsion-injective left approximation.

Proof. (1)⟺(2) It is guaranteed by our hypotheses.

(1)⟺(3) The proof for modules ([15, Proposition 4.1] is also valid here.

The weak version of (4) implies (2), also by adapting [15, Proposition 4.1] It only remains to prove that (1)-(2) imply the weak version of (4) and, by

our previous corollary, the strong version will hold as well. Indeed, (1) implies that the inclusion functor i : T ⟶ G also has a left adjoint r : G ⟶ T . If now M ∈ Ob(G), we consider an injective envelope r(M) ⟶^{λ} I in T and the composition M ⟶^{ϕ} i ∘ r(M) ⟶^{i(λ)} i(I) is the desired left _{T} Inj-approximation, where ϕ is the adjoint map.

Example 2.6. (1) If T ⊆ Ob(G) is a localizing subcategory closed under taking injective envelopes, then the corresponding torsion-injective objects are exactly the objects of G which are torsion and injective. In this situation, the above corollary is telling us that every object of G has a (minimal) torsion and injective left approximation iff T is closed under direct products. Hence, over a commutative integral domain which is not a field, not every module has a torsion and injective left approximation.

(2) If M is any module, then every R-module has a M-injective M-generated (minimal) left approximation if, and only if, M^{I} is M-subgenerated (i.e. embeddable in some quotient of a direct sum of copies of M), for every index set I. Indeed, one easily sees that the class σ[M] of M-subgenerated modules is closed for products iff M^{I} ∈ σ[M], for every I. Now our claim follows from the above corollary and [18, 3.16.3].

3. Product-complete modules

Given any R-module M we denote by Add M the full subcategory of R-modules which are direct summands of coproducts of copies of M. Note that Add M is the smallest full subcategory of Mod R which contains M and is closed under forming arbitrary coproducts and direct summands. Our main result is a characterization of the fact that Add M is closed under taking products.

Theorem 3.1. The following are equivalent for an R-module M:

(1) Add M is closed under taking products.

(2) Every product of copies of M is a direct summand of a coproduct of copies of M.

(3) Every product of copies of M is a coproduct of (indecomposable) direct summands of M.

(4) Every R-module has a (minimal) left Add M-approximation.

We call a module M product-complete if M satisfies the equivalent conditions of the preceding theorem.

Recall that a module M is Σ-pure-injective if every coproduct of copies of M is pure-injective. We shall work with the following characterization of a Σ-pure-injective module.

Lemma 3.2. A module M is Σ-pure-injective if and only if there exists a cardinal κ such that every product of copies of M is a pure submodule of a coproduct of modules having cardinality at most κ.

Proof. See [11].

We shall also need the following property of a Σ-pure-injective module.

Lemma 3.3. Any Σ-pure-injective module is a coproduct of indecomposable modules with local endomorphism ring.

Proof. See [11, 19].

Proof of Theorem 3.1. (1) ⇒ (2) Clear.

(2) ⇒ (3) M is Σ-pure-injective by Lemma 3.2. It follows from Lemma 3.3 in combination with the Krull-Remak-Schmidt-Azumaya Theorem that every direct summand of a coproduct of copies of M decomposes into indecomposable modules which are direct summands of M. Thus (2) implies (3).

(3) ⇒ (4) M is Σ-pure-injective by Lemma 3.2. Thus Add M consists of pureinjective modules and is closed under taking products. It follows from Corollary 2.3 that every R-module has a minimal left Add M-approximation.

(4) ⇒ (1) Use Lemma 2.1.

We give an example of a Σ-pure-injective module which is not product-complete.

Example 3.4. Let R = ℤ and let M = ℤ_{p^{∞}} be a Prüfer group. This module is Σ-pure-injective but not product-complete. However M ∐ ℚ is product-complete.

We list some basic properties of product-complete modules.

(1) The class of product-complete modules is closed under finite coproducts.

(2) Let I be any non-empty set. Then a module M is product-complete if and only if M^{(I)} is product-complete.

(3) If M is product-complete, then M^{I} is product-complete for every set I.

(4) A module M is Σ-pure-injective if and only if there exists a non-empty set I such that M^{I} is product-complete. In fact, if M is Σ-pure-injective, then one chooses I big enough such that for any set J any indecomposable direct summand of M^{J} already occurs as a direct summand of M^{I}.

We have the following categorical characterization of product-complete modules.

Proposition 3.5. An R-module M is product-complete if and only if there exists a (locally noetherian) Grothendieck category A and a functor F: A → Mod R which commutes with products and coproducts, such that F induces an equivalence between the full subcategory Inj A of injective objects in A and Add M.

Proof. We use the functor category D(R) = (mod R^{op}, Ab). Suppose that M is product-complete and consider the full subcategory T of all objects X in D(R) with Hom(X, M ⊗_{R} −) = 0. This is a localizing subcategory and we can therefore form the quotient category A = D(R)/T , e.g. see [8]. The composition of the embedding Mod R → D(R), N ↦ N ⊗_{R} −, with the quotient functor D(R) → A identifies Add M with Inj A since M ⊗_{R} − is an injective cogenerator for A, and therefore A is locally noetherian since Inj A is closed under forming coproducts [16]. The composition of the right adjoint of the quotient functor with the functor D(R) → Mod R, X ↦ X(R), gives the functor F: A → Mod R which has the desired properties. Conversely, suppose there exists a functor F: A → Mod R which commutes with products and coproducts, such that F induces an equivalence Inj A → Add M. It follows that Add M is closed under taking products since Inj A is closed under taking products, and therefore Add M is product-complete.

Corollary 3.6. If M is product-complete, then Add M is closed under taking direct limits.

Proof. If A is a locally noetherian category, then Inj A is closed under taking direct limits [8, p.358]. The assertion is therefore a consequence of the construction given in the preceding proof.

In [20], Ziegler introduced a topology on the set Ind R of isomorphism classes of indecomposable pure-injective R-modules, and it is also possible to characterize the product-complete modules in terms of this topology.

Proposition 3.7. A module M is product-complete if and only if M is Σ-pure-injective and the indecomposable direct summands of M form a Ziegler-closed subset of Ind R.

Proof. In [13, Proposition 8.7] it is shown that for every Σ-pure-injective module M the indecomposable direct summands of products of copies of M form a Zieglerclosed set. Using the characterization (2) in Theorem 3.1 the assertion follows.

We now characterize the finitely generated modules which are product-complete.

Proposition 3.8. A finitely generated module M is product-complete if and only if S = End_{R}(M) is left coherent and right perfect, and M is finitely presented as S^{op}-module.

Proof. The functors Hom_{R}(M, −) and −⊗_{S}M induce mutually inverse equivalences between Add M and the full subcategory Proj S of projective S-modules since M is finitely generated. In particular, the canonical map ε_{N}: Hom_{R}(M, N) ⊗_{S} M → N is an isomorphism for all N in Add M. Now suppose that M is product-complete. Since M^{I} ∈ Add M, we get that S^{I} ∈ Proj S, for every set I. A classical result of Chase says that S is right perfect and left coherent [5, Theorem 3.3]. Moreover, since ε_{M^{I}} is an isomorphism for every set I, due to our above observation, the canonical map S^{I} ⊗_{S} M → M^{I} is an isomorphism, for every set I. This is wellknown to be equivalent to the fact that M is finitely presented as S^{op}-module; see, e.g., [17, Lemma I.13.2]. To prove the converse, we need to check that M^{I} ∈ Add M for every set I. We have S^{I} ∈ Proj S since S is left coherent and right perfect, and S^{I} ⊗_{S} M ≃ M^{I} since M is finitely presented over S^{op}. Thus M^{I} ∈ Add M, and therefore M is product-complete.

Remark 3.9. (1) When M = R in the above proposition, we get that every Rmodule has a minimal projective left approximation if and only if R is right perfect and left coherent [2, Proposition 3.5].

(2) The hypothesis that M is finitely generated cannot be omitted in the above proposition. Indeed, if R is a right noetherian ring and M is a coproduct of a representative set of indecomposable injective R-modules, then Add M = Inj R, so that M is product-complete. However, in general, M has an infinite number of direct summands and so S = End_{R}(M) is not right perfect.

4. Endofinite modules

Recall that an R-module M is endofinite if it has finite length as module over its endomorphism ring End_{R}(M)^{op}. In this section we discuss the relation between endofinite and product-complete modules. The basic result is the following.

Theorem 4.1. A module M is endofinite if and only if every direct summand of M is product-complete.

Proof. Suppose first that M is indecomposable. It is a well-known result due to Gruson [10] and Garavaglia [9] that M is endofinite if and only if Add M is closed under taking products. We refer to [14, Corollary 10.5] for a proof since [10] contains no proof and [9] has never been published. Now suppose that M is an arbitrary module. We use the fact that M is endofinite if and only if there are indecomposable endofinite modules M_{1}, …, M_{n} and sets I_{1}, …, I_{n} such that M ≃ ∐_{i=1}^{n} M_{i}^{(I_{i})}; see [6, Proposition 4.5]. The fact that every endofinite module is product-complete is now a consequence of the basic properties of product-complete modules listed after Theorem 3.1, and the fact that every indecomposable endofinite module is product-complete. Conversely, let M be a module such that every direct summand is product-complete. It follows from Proposition 3.7 that the indecomposable direct summands of M form a Ziegler-closed subset U of Ind R having the property that every subset V ⊆ U is also Ziegler-closed. However, Ind R is a quasi-compact space [20, Theorem 4.9] and therefore U needs to be finite. Thus M is endofinite since each N ∈ U is endofinite and M is a coproduct of modules in U by Theorem 3.1. This completes the proof.

We fix an R-module M with S = End_{R}(M)^{op}. Recall that an additve subgroup of M is of finite definition if it arises as the kernel of a morphism M → M ⊗_{R} X, m ↦ m ⊗ x, for some finitely presented R^{op}-module X, and some element x ∈ X. Note that a subgroup of finite definition is a S-submodule of M. If M is Σ-pure-injective, then any finitely generated S-submodule of M is a subgroup of finite definition. We shall now prove that every subgroup of finite definition is a finitely generated S-module provided that M is product-complete.

Proposition 4.2. A module M is product-complete if and only if the following conditions hold:

(1) M has the descending chain condition on subgroups of finite definition.

(2) Every subgroup of finite definition is a finitely generated End_{R}(M)^{op}-module.

Proof. We use again the embedding Mod R → D(R), N ↦ T_{N} = N ⊗_{R} −. Identifying M = Hom(T_{R}, T_{M}), any additive subgroup U is of finite definition if and only if U = Hom(T_{R}/F, T_{M}) for some finitely generated subfunctor F ⊆ T_{R}. Note that U is finitely generated as S-module if and only if there exists a map ϕ: T_{R}/F → T_{M^{n}} for some n ∈ ℕ such that every map T_{R}/F → T_{M} factors through ϕ. Now suppose that M is product-complete. Then M is Σ-pure-injective and it is well-known that (1) holds [11, 19]. To check (2) let U = Hom(T_{R}/F, T_{M}) be a subgroup of finite definition. There exists a map ϕ: T_{R}/F → T_{M^{I}} for some set I such that every map T_{R}/F → T_{M} factors through ϕ. We find a set J such that M^{I} is a direct summand of M^{(J)}, and the map T_{R}/F → T_{M^{I}} → T_{M^{(J)}} factors through T_{M^{(J′)}} for some finite subset J′ ⊆ J since F/T_{R} is finitely generated. Denoting by ϕ′: T_{R}/F → T_{M^{(J′)}} the corresponding map it is clear that every map T_{R}/F → T_{M} factors through ϕ′. Thus U is a finitely generated S-module and (2) holds. Using similar arguments it is not hard to check that M is product-complete if (1) and (2) hold. This is left to the reader.

Lemma 4.3. M is endofinite if and only if M is Σ-pure-injective and a noetherian module over S.

Proof. It is well-known that M is Σ-pure-injective if and only if M has the dcc on subgroups of finite definition [11, 19]. Thus every endofinite module is Σ-pure-injective since every subgroup of finite definition of M is a S-submodule. Conversely, if M is Σ-pure-injective, then every finitely generated S-submodule is a subgroup of finite definition [6, Lemma 4.1]. It follows that M is artinian over S since every S-submodule of M is finitely generated if M is noetherian over S. Thus M has finite endolength.

Corollary 4.4. Let M be an R-module and suppose that End_{R}(M) is left noetherian. Then M is endofinite if and only if M is product-complete.

Proof. If M is product-complete, then M is finitely generated over S = End_{R}(M)^{op} by Proposition 4.2. Thus M is noetherian over S and the assertion follows from Lemma 4.3.

The semi-simple modules form another class of modules where endofinite and product-complete modules coincide.

Proposition 4.5. For a semi-simple R-module M the following are equivalent: (1) M is endofinite.

(2) M is product-complete.

(3) Every product of copies of M is semi-simple.

(4) The ring R/ann M is semi-simple.

Proof. (1) ⇒ (2) Use Theorem 4.1.

(2) ⇒ (3) Clear.

(3) ⇒ (4) Consider the canonical map R → M^{M} which induces a monomorphism R/ann M → M^{M}. Any submodule of a semi-simple module is semi-simple, and therefore R/ann M is a semi-simple ring.

(4) ⇒ (1) Any module over a semi-simple ring is endofinite.

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Theorem 4.6. A module M is endofinite if, and only if, every direct summand of M is product-complete

Proof. (The initial part of a proof without Ziegler spectrum?) By using a result of Garavaglia ([9], but see [14] for a proof) and our Theorem 3.1, we know that an indecomposable module is endofinite iff it is product-complete. With that in mind, he ”only if” part is a consequence of [6, Proposition 4.5] and the listed properties of product-complete modules.

For the ”if” part, it is not restrictive to assume that M = ∐_{i∈I} M_{i}, where the M_{i} are pairwise nonisomorphic indecomposable, each of which must be endofinite by our above observation. Let us fix any index i ∈ I and, for each subset J ⊆ I with ι ∉ J, take H_{J} = {x ∈ M_{i} : f(x) = 0,for every f ∈ Hom_{R}(M_{i}, M_{j}), with j ∈ J}. One readily sees that H_{J} is a R-S_{i}-subbimodule of M_{i}, where S_{i} = End_{R}(M_{i}). Then the family { H_{J} : J ⊆ I, i ∉ J, J finite } has a minimal element which is easily seen to be unique. We choose a finite subset J_{1} ⊆ I, with i ∉ J_{1}, such that H_{J_{1}} is that minimal element. Now we take the family { H_{J} : J ⊆ I, i ∉ J, J ∩ J_{1} = ∅,J finite }, which must have also a minimal element, say H_{J_{2}}. By proceeding in this way, we get an ascending chain H_{J_{1}} ⊆ H_{J_{2}} ⊆... os S_{i}-submodules of M_{i}, with the J_{n} finite and pairwise disjoint. Then there is a m ≥ 1 such that H_{J_{m}} = H_{J_{m+1}}=... Put H=H_{J_{m}} and we claim that H=M_{i}, thus implying that, for all but finitely many indices j ∈ I, Hom_{R}(M_{i}, M_{j}) = 0. Indeed, M_{i}/H embeds in a direct product of copies of M_{J_{m}} = ∐_{j∈J_{m}} M_{j} and in a direct product of copies of M_{J_{m+1}} = ∐_{j∈J_{m+1}} M_{j}. By looking at the minimal left Add(M)-approximation E of M_{i}/H and bearing in mind that M_{J_{m}} and M_{J_{m+1}} are both product-complete, we have that E ∈ Add(M_{Jm+1}) ∩ Add(M_{J_{m}}) = 0 and then M_{i}/H = 0, as we claimed.

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

   E-mail address: henning@mathematik.uni-bielefeld.de

Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo - MU, Spain

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