En “Rings, Hopf algebras and Brauer groups”. Marcel–Dekker, 1998, 147–157

Decomposition of locally finite modules

J. M. García

Department of Applied Math.

P. Jara^{∗}, L. M. Merino^{∗}

Department of Algebra

University of Granada 18071–Granada

Abstract

This paper deals with the decomposition theory of locally finite modules in indecomposable submodules. Applications to the structure of coalgebras and comodules are also provided.

0. Introduction.

One of the most beautiful results in ring theory are the decomposition theorem of modules. The first of them is the primary decomposition theorem of modules over a principal ideal domain. There are many extensions of this result to different kinds of rings. In this paper we deal with the structure theorems for locally finite modules, i. e., finitely generated submodules are finite–dimensional. We will obtain a very general like primary decomposition theorem for this class of modules.

In the commutative case the primary components are associated to prime ideals; the reason is that there is not any link between prime ideals. In the non commutative case the situation is completely different. For instance, we get a very simple case when we study modules over finite–dimensional algebras; in that case we can not distinguish between two prime ideals P and Q satisfying QP ⊂ Q ∩ P, so we say that there is a link from Q to P. This is equivalent to the existence of a non splitting short exact sequence 0 → T → X → S → 0 with T, S simple modules and Ann(T) = Q, Ann(S) = P. The existence of a link is related with the inner structure of the injective hull E(T) of T, i. e., if there is a link from Q to P, then there exists a non zero homomorphism from E(T) to S. Therefore, if we study modules using injective modules as a tool (or equivalently hereditary torsion theories), then P and Q are indistinguishable.

∗Research partially supported by DGICYT (PB94–0791)

With these facts in mind, it is natural to use big primes to define primary decomposition instead of single prime ideals. These big primes will be the connected components of a graph, the graph with vertices the prime ideals and arrows the link relations between prime ideals.

The core of this paper is the study of a locally finite module M. To do that we use the smallest Grothendieck subcategory σ[M] of R –Mod containing M, see [7]. So following [3], we define a set of prime ideals associated to M and using the simple modules in σ[M] we define links between these prime ideals. Every connected component defines a submodule of M: the primary component, and also we prove that M is the direct sum of its primary components. In order to perform the theory we define indecomposable modules. First σ–indecomposable, see the definition in the text, and later indecomposable using the two sided structure of M as left R–module and as right End(_{R}M)^{opp}–module. As a consequence, taking into account that for any locally finite module M there exists a module C ∈ σ[M] such that σ[C] = σ[M] and C is self–injective (i. e., injective in σ[C]) and self–cogenerator (i. e., cogenerator of σ[C]), we study how the decomposition of C and M influences on the decomposition of the category σ[C] = σ[M] and conversely. In that way we prove that the case self–injective and self–cogenerator is particularly interesting specially because on these modules we have that both afore–mentioned indecomposition notions coincide.

At that point important examples of modules such that M is self–injective and self–cogenerator arise in the literature. For instance, if R is a K–algebra, then the dual coalgebra R^{∘} has the property that σ[_{R}R^{∘}] is the subcategory of all locally finite modules, and it is isomorphic to the category of right R^{∘}–comodules and also _{R}R^{∘} is self–injective and self–cogenerator. So as a byproduct a decomposition theory for all locally finite modules is obtained. This decomposition provides also a decomposition of R^{∘} as coalgebra.

A parallel theory can be performed in the case we take a coalgebra C. In that case the category of right C–comodules is isomorphic to the subcategory of all rational left C^{∗}–modules, which can be also described as σ[_{C^{∗}}C]. The decomposition of C provides a decomposition as a direct sum of indecomposables coalgebras.

In both cases a decomposition theory of comodules can be performed. In the literature a theory of decomposition of comodules was developed in [8]. We include in this paper an example that provides a counter–example to the decomposition theory studied in [8]. The main point to built this example was to taking into account that on the contrary to the coalgebra case, a comodule M not necessarily is self–injective and self–cogenerator. So the category σ[M] may have simple modules non embedded in M.

1. Closed subcategories and uniform filters.

Let R be a non necessarily commutative ring, and let M be a left R– module. We denote by σ[M] the full subcategory of R –Mod consisting of all submodules of M–generated R–modules. It is well known that σ[M] is a Grothendieck category. The family of all cyclic submodules of M is a system of generators of σ[M]. On the other hand σ[M] is closed under taking submodules, quotient modules and direct sums, hence it is a closed subcategory of R –Mod. Indeed σ[M] is the smallest closed subcategory containing M.

Let R be a K–algebra. A left R–module M is called locally finite if every element m ∈ M generates a finite dimensional (as K–vectorial space) submodule. It is clear that submodules and quotient modules of locally finite modules are locally finite; also the direct sum of locally finite modules is locally finite. As a consequence the full subcategory of locally finite modules is another example of a closed subcategory of R –Mod.

Closed subcategories in R –Mod, so in σ[M], may be parametrized by families of left ideals of R in the following way. Let C be a closed subcategory of R –Mod, we define:

V_{C} = {I ⊆_{R} R: R/I ∈ C}.

1 Lemma. With the above notation V_{C} satisfies the following properties:

(a) Let I ∈ V_{C} and let J be a left ideal such that I ⊆ J, then J ∈ V_{C};

(b) Let I, J ∈ V_{C}, then I ∩ J ∈ V_{C};

(c) Let I ∈ V_{C} and a ∈ R, then (I: a) ∈ V_{C}.

A family of left ideals V satisfying the properties (a), (b), (c) in the above Lemma is called a uniform or linear filter in R. And we can prove that there exists a one–to–one correspondence between closed subcategories in R –Mod and uniform filters in R.

Intersection of uniform filters is a uniform filter. But, in general, the union of uniform filters is not a uniform filter. Nevertheless we can define the join of a family of uniform filters as the smallest uniform filter containing every filter in the family, i. e., the intersection of all uniform filters containing all the filters in the family. In the above correspondence intersection (resp. join) of uniform filters corresponds to the intersection (resp. join) of closed subcategories.

2 Example. Let A be an ideal^{1} of R, then η(A) = {I ⊆_{R} R: A ⊆ I} is a uniform filter in R. Indeed η(A) is the smallest uniform filter containing A.

3 Example. Let {P_{α}: α ∈ Λ} be a non–empty family of prime ideals of R, then

∨_{α}η(P_{α}) = {I ⊆_{R} R: there exist α_{1}, …, α_{t} ∈ Λ such that
P_{α_{1}} ∩ ⋯ ∩ P_{α_{t}} ⊆ I}

is the smallest uniform filter containing {P_{α}: α ∈ Λ}.

4 Example. Let M be a left R–module, then σ[M] is a closed subcategory of R –Mod. We denote by η(M) the uniform filter associated to σ[M].

2. Simple modules in σ[M].

In the following, in this paper, M will be a locally finite left R–module. In order to perform the theory of decomposition of M we will relate simple modules in σ[M] and prime ideals of R.

First of all we recall that there exists a one–to–one correspondence between maximal cofinite ideals and simple finite–dimensional left R–modules.

This correspondence can be particularized to σ[M] in the following sense. Let S be a simple module in σ[M], since cyclic submodules of M constitute a system of generators of σ[M], then there exists m ∈ M and an epimorphism Rm → S → 0. Hence Ann(S) ⊇ Ann(Rm). It is clear that Ann(S) is a maximal cofinite ideal of R.

On the other hand, if P is a maximal cofinite ideal of R such that P ⊇ Ann(Rm) for some m ∈ M, then since Rm is of finite dimension, it has a composition series, so there exist maximal cofinite ideals P_{1}, …, P_{t} such that P_{1} ⋯ P_{t} ⊆ Ann(Rm) ⊆ P. Now P is prime, so there exists an index i such that P_{i} ⊆ P, and since both, P_{i} and P, are maximal, we have P = P_{i}. Hence there exists a simple submodule S in σ[M] such that Ann(S) = P.

1As usual ideal means two–sided ideal.

5 Lemma. There exists a one–to–one correspondence between maximal cofinite ideals containing Ann(Rm) for some m ∈ M and isomorphic classes of simple modules in σ[M].

Let us call Spec(M) a representative set of isomorphism classes of simple modules in σ[M] and Spec_{M}(R) the image of Spec(M) in Spec(R) by the above correspondence.

3. Links.

Before to follow with these arguments to find a decomposition theory of a locally finite left R–module M, we need some background.

Let N ∈ σ[M], if N is injective in R –Mod, then N is injective in σ[M]. On the other hand, for any injective left R–module X if we define r_{η(M)}(X) the largest submodule of X contained in σ[M], then r_{η(M)}(X) is injective in σ[M]. As a consequence, if E(N) is the injective hull of N in R –Mod, then we can prove that r_{η(M)}(E(N)) is the injective hull of N in σ[M]. Let us denote r_{η(M)}(E(N)) simply by E_{M}(N).

We now proceed to define links between elements in Spec(M). Let S, T be simple modules in σ[M], we say that there is an injective link from T to S in σ[M] if there exists a non–zero homomorphism f ∈ Hom_{R}(E_{M}(T), E_{M}(S)). So we get a graph when we take the isomorphism classes of simple modules in σ[M] as vertices and the above defined injective links arrows.

The graph of simple modules can be implemented also using another link relation and actually using maximal cofinite ideals. To do that we prove the following results:

Let S and T be simple modules in σ[M], we say that there exists an Ext– link from T to S in σ[M] if there exists a non–splitting short exact sequence 0 → T → X → S → 0 in σ[M]. We represent it simply by T ⇝_{M} S.

6 Proposition. Let S be a simple module in σ[M], then there exists a simple module T ⊆ M

such that T ⇝_{M} ⋯ ⇝_{M} S.

Proof. Let S ∈ σ[M] be a simple module, then there exists m ∈ M and an epimorphism Rm → S → 0. Since Rm is of finite dimension, there exists a composition series

0 = N_{0} ⊂ ⋯ ⊂ N_{t} = Rm

such that N_{t}/N_{t−1} ≅ S. Let us call S_{i} = N_{i}/N_{i−1}, i = 1, …, t, the composition factors of Rm. If S_{i−1} is not linked with S_{i}, then we have an splitting short exact sequence

0 → S_{i−1} → N_{i}/N_{i−2} → S_{i} → 0

So there exists N′_{i−1} ⊆ N_{i} such that N_{i}/N_{i−2} = N_{i−1}/N_{i−2} ⊕ N′_{i−1}/N_{i−2} and N′_{i−1}/N_{i−2} ≅ S_{i}. So we can change the above composition series to the next one

0 = N_{0} ⊂ ⋯ ⊂ N_{i−2} ⊂ N′_{i−1} ⊂ N_{i} ⊂ ⋯ ⊂ N_{t} = Rm

Indeed in that way we may prove that there exists a chain of Ext–links in σ[M] from N_{1} to S (or N_{1} is isomorphic to S). □

7 Proposition. Let S, T simple modules in σ[M] such that there exists a Ext–link T ⇝_{M} S in

σ[M], then there exist an injective link from T to S in σ[M]. Conversely, if S and T are simple modules in σ[M] and there exists an injective link from T to S in σ[M], then there exists a chain of Ext–links in σ[M] starting in T and finishing in S.

Proof. The first part is clear. Let us assume that there exists an injective link from T to S in σ[M] and let 0 ≠ f ∈ Hom_{R}(E_{M}(T), E_{M}(S)). So f(E_{M}(T)) ∩ S ≠ 0, hence S ⊆ f(E_{M}(T)). Since S is simple, there exists m ∈ E_{M}(T) such that f(Rm) ≅ S. As T is simple and essential in E_{M}(T) we have T ⊆ Rm, so there exists a composition series of Rm

0 = N_{0} ⊂ ⋯ ⊂ N_{t} = Rm

such that N_{1} = T and N_{t}/N_{t−1} ≅ S. Then an argument like in the proof of Proposition 6 gives the announced result. □

8 Proposition. Let M be a left R–module, then the connected components in the graphs on Spec(M) with the injective link relation and Spec(M) with the Ext–link relation coincide.

Let P and Q be maximal cofinite ideals in R, we say that there exists a link from Q to P if QP ⊂ Q ∩ P. Thus we may define a graph with vertices the maximal cofinite ideals in R and an arrow from Q to P if there exists a link from Q to P. It is well known, even in the non noetherian case, that there exists a link from Q to P if and only if there exists a non–splitting short exact sequence 0 → T → X → S → 0, see [2]. If there exists a link from P to Q we represent it by Q ⇝ P.

We obtain that if P, Q are maximal cofinite ideals such that P = Ann(S) and Q = Ann(T), where S, T are simple left R–modules, then Q ⇝ P if and only if there exists an Ext–link from T to S in R –Mod, i. e., T ⇝ S.

We may define the links relative to M between maximal cofinite ideals in Spec_{M}(R). With the above notation, if P, Q ∈ Spec_{M}(R), then Q ⇝_{M} P if

T ⇝_{M} S. Then we have a graph built on Spec_{M}(R), which is isomorphic to

the graph on Spec(M) given by the Ext–link relation in σ[M].

4. Decomposition of modules.

Let P ∈ Spec_{M}(R), then we may consider the uniform filter η(P), and C_{η(P)}, the associated closed subcategory. For any N ∈ σ[M] let us call r_{η(P)}(N) the largest submodule of N contained in C_{η(P)}, i. e., r_{η(P)}(N) = {n ∈ N: P n = 0}.

If we consider a set X of elements in Spec_{M}(R) we may proceed in the same way, i. e., η(X) will be the least uniform filter containing X. For any module N ∈ σ[M] we may define

r_{η(X)}(N) = {n ∈ N: there exists P_{1}, …, P_{t} ∈ X such that
(P_{1} ∩ ⋯ ∩ P_{t})n = 0},

see Example 3. We will apply this when X is a connected component of Spec_{M}(R).

9 Proposition. Let M be a left R–module and Spec_{M}(R) = ∪{X_{α}: α ∈ Λ} the decomposition of Spec_{M}(R) in connected components. Let N ∈ σ[M], then the family of submodules {r_{η(X_{α})}(N): α ∈ Λ} is independent.

Proof. Let X, Y be connected components such that X ≠ Y, then r_{η(X)}(N) ∩ r_{η(Y)}(N) = 0. Let n ∈ r_{η(X)}(N) ∩ r_{η(Y)}(N), then there exists P_{1}, …, P_{t} ∈ X and Q_{1}, …, Q_{s} ∈ Y such that (P_{1} ∩ ⋯ ∩ P_{t})n = 0 = (Q_{1} ∩ ⋯ ∩ Q_{s})n. Since Rn is finite–dimensional, then it is of finite length and so there exist prime ideals P_{i} and Q_{j} such that P_{i} = Q_{j}, which is a contradiction. □

In general the quotient module N/r_{η(X)}(N) do not satisfy the following relation r_{η(X)}(X/r_{η(X)}(N)) = 0. In order to arrange this obstacle we define the product of uniform filter. Let V_{1} and V_{2} uniform filters in R, we define a new uniform filter V_{1} V_{2} in the following way.

V_{1} V_{2} = {I ⊆_{R} R: there exists K ∈ V_{2} such that I ⊆ K and (I: a) ∈ V_{1}

for every a ∈ K}

In a similar way we may define the product of closed subcategories C_{1} and C_{2} of R –Mod as follows:

C_{1} C_{2} = {X ∈ R –Mod: there exists a short exact sequence
0 → C_{1} → X → C_{2} → 0 such that C_{i} ∈ C_{i}}

It is an exercise to prove that C_{V_{1}} C_{V_{2}} = C_{V_{1} V_{2}}.

If V_{i}, i = 1, 2, are uniform filters in R such that C_{V_{1}} ⊆ σ[M], it may be happen that the closed subcategory C_{V_{1} V_{2}}, associated with V_{1} V_{2} is not contained in σ[M]. So in that case we may define a restricted product, denoted V_{1} · V_{2}, which is the intersection of V_{1} V_{2} with η(M). The reader can prove that

C_{V_{1} · V_{2}} = {X ∈ σ[M]: there exists a short exact sequence
0 → C_{1} → X → C_{2} → 0 such that C_{i} ∈ C_{i}}

Hence we may define a restricted product of closed subcategories C_{1} and C_{2} of σ[M]

C_{1} · C_{2} = {X ∈ σ[M]: there exists a short exact sequence
0 → C_{1} → X → C_{2} → 0 such that C_{i} ∈ C_{i}}

It is obvious that C_{V_{1} · V_{2}} = C_{V_{1}} · C_{V_{2}}.

A uniform filter V ≤ η(M) is called idempotent or a Gabriel filter in σ[M] if V · V = V. It is very easy to prove that idempotent uniform filters V ≤ η(M) correspond one–to–one to hereditary torsion classes in σ[M] in the sense of [6].

We can iterate the product of uniform filters. Let us define V^{n} = V^{n−1} · V = V · V^{n−1} and V^{∞} = ∪_{n} V^{n}. Then V^{∞} is a uniform filter.

Let V be a uniform filter such that V ≤ η(M) with associated closed subcategory C_{V}, then we define:

Z(V) = Z(C_{V}) = {P ∈ Spec_{M}(R): P ∈ V}
K(V) = K(C_{V}) = {P ∈ Spec_{M}(R): V(R/P) = 0}

10 Lemma. If V is a symmetric uniform filter (i. e., V has a cofinal set consisting of two–sided ideals), then {Z(V), K(V)} is a partition of Spec_{M}(R).

11 Lemma. Let V_{1}, V_{2} be uniform filters, then

Z(V_{1} V_{2}) = Z(V_{1} ∨ V_{2}) = Z(V_{1}) ∪ Z(V_{2})

As a consequence for any uniform filter V we have Z(V^{∞}) = Z(V).

Now it is possible to extend Proposition 9 to the uniform filters η(X_{α})^{∞}. In fact we have:

12 Proposition. Let M be a locally finite left R–module and Spec_{M}(R) = ∪{X_{α}: α ∈ Λ} a decomposition of Spec_{M}(R) in connected components. Let N ∈ σ[M], then the family of submodules {r_{η(X_{α})^{∞}}(N): α ∈ Λ} is independent and N is the direct sum N = ⊕{r_{η(X_{α})^{∞}}(N): α ∈ Λ}.

Proof. Let n ∈ N, then Rn is finite–dimensional, so I = Ann(Rn) is a cofinite ideal of R and R/I is a finite–dimensional ring. Since Rn is a finitely generated R/I–module, then Rn = ⊕_{β}{r_{η(Y_{β})^{∞}}(Rn): β ∈ B}, where {Y_{β}: β ∈ B} is a partition in connected components of Spec(R/I). Hence Rn ⊆ ⊕_{α}r_{η(X_{α})^{∞}}(N). □

The following is a useful consequence of this result.

13 Lemma. For any connected component X of Spec_{M}(R) the uniform filter η(X)^{∞} is idempotent in σ[M].

Proof. If η(X)^{∞} · η(X)^{∞} ≠ η(X)^{∞} there exists N ∈ σ[M] such that N = r_{η(X)^{∞}·η(X)^{∞}}(N) ≠ r_{η(X)^{∞}}(N). Using the decomposition given in Proposition 12, there exists a connected component X_{α} such that X_{α} ≠ X and r_{η(X)^{∞}}(N) ≠ 0. So r_{η(X_{α})}(N) ≠ 0 and there exists a simple submodule S ⊆ N such that Ann(S) ∈ X_{α}. Then we have:

Ann(S) ∈ Ass(M) ⊆ Z(η(X)^{∞} · η(X)^{∞}) = Z(η(X)) = X,

which is a contradiction. □

At this point we may ask on the following question: Under what conditions we have that V^{∞} is idempotent in σ[M]? Remember that if R is left noetherian, then for any uniform filter V in R –Mod we have that V^{∞} is idempotent in R –Mod.

14 Proposition. Let L be an idempotent uniform filter in σ[M] and let N ∈ σ[M]. If Ass(N) ⊆ Z(L), and Z(L) ⊆ Spec_{M}(R) is link closed, then N ∈ C_{L}.

Proof. Let n ∈ N, then Rn is of finite dimension and there exist P_{1}, …, P_{m} maximal cofinite ideals of R such that P_{1} ⋯ P_{m}n = 0. Since Z(L) is link closed we have P_{1}, …, P_{m} ∈ L, hence P_{1} ⋯ P_{m} ∈ L. Therefore, since P_{1} ⋯ P_{m} ⊆ Ann(n), we have Ann(n) ∈ L and N ∈ C_{L}. □

15 Corollary. Let L be a idempotent uniform filter in σ[M]. If Z(L) ⊆ Spec_{M}(R) is link closed, then C_{L} ⊆ σ[M] is closed under essential extensions.

Proof. Let N′ ⊆^{e} N an essential extension and let N′ ∈ C_{L}, then Ass(N′) = Ass(N) ⊆ L. Hence N′ ∈ C_{L}. □

5. Decomposable modules.

As an application every module N in σ[M] can be decomposed as a direct sum of submodules. We are now interesting in knowing extra properties of this decomposition. Let us study it in detail.

If we apply this decomposition to M itself, then, to simplify notation, let us assume the decomposition is M = ⊕_{α}M_{α}, where M_{α} = r_{η(X_{α})^{∞}}(M).

The family {M_{α}: α ∈ Λ} determines the partition of Spec(M) in its connected components.

Let us consider σ[M_{α}] ⊆ σ[M]. It is a closed subcategory, and it is easily proved that if α ≠ β then σ[M_{α}] ∩ σ[M_{β}] = {0}.

What relationship there exists between σ[M_{α}] and C_{η(X_{α})^{∞}}? It is very easy to prove the inclusion σ[M_{α}] ⊆ C_{η(X_{α})^{∞}}. The other inclusion is not true in general unless σ[M_{α}] was closed under extensions, i. e., η(M_{α}) was an idempotent uniform filter in σ[M].

A locally finite left R–module M is called σ–decomposable if:

(1) M = N_{1} ⊕ N_{2}, with N_{1}, N_{2} non zero submodules;

(2) σ[N_{1}] ∧ σ[N_{2}] = {0};

(3) Z(σ[N_{1}]) ∪ Z(σ[N_{2}]) = Spec_{M}(R); (4) For any simple module X in Z(σ[N_{i}]) and for any simple module Y in

Z(σ[N_{j}]) if i ≠ j we have Hom_{R}(E_{M}(X), Y) = 0.

As a consequence of property (4) each connected component of Spec_{M}(R) is contained only in one of the Z(σ[N_{i}]), i = 1, 2.

The underlying idea is that a σ–decomposable module M can be decompose due to there are submodules N_{1} and N_{2} satisfying M = N_{1}⊕N_{2} and such that σ[N_{1}] and σ[N_{2}] yield a partition of Spec_{M}(R) which respect the connected components.

16 Theorem. Let M be a locally finite left R–module such that M = N_{1} ⊕ N_{2} with N_{1} and N_{2} non zero submodules, then the following statements are equivalent:

(a) M = N_{1} ⊕ N_{2} is a σ–decomposition of M;

(b) For any N ∈ σ[M] we have N = τ_{1}(N) ⊕ τ_{2}(N), where τ_{i} = r_{σ[N_{i}]^{∞}} and Z(σ[N_{1}]), Z(σ[N_{2}]) are link closed in σ[M].

We say that a locally finite left R–module M is σ–indecomposable if is not σ–decomposable. As a consequence we have that M is σ–indecomposable if and only if Spec_{M}(R) is connected.

Let M be a locally finite left R–module, we define M to be indecomposable if when M = N_{1} ⊕ N_{2} as (R, End(_{R}M)^{opp})–bimodules, then either M = N_{1} or M = N_{2}. Our next result studies a particular case in which the decomposition given in Theorem 16 has indecomposable components, i. e., when σ–indecomposable modules are indecomposable.

17 Theorem. Let M be a locally finite left R–module such that M is an injective cogenerator of σ[M]. If M = N_{1} ⊕ N_{2} is a decomposition de M then the following statements are equivalent:

(a) M = N_{1} ⊕ N_{2} as (R, End(_{R}M)^{opp})–bimodules;

(b) M = N_{1} ⊕ N_{2} is a σ–decomposition.

Proof. Since M is an injective cogenerator, every simple module in σ[M] can be embedded in M.

(a) ⇒ (b). Since Ass(M) = Spec_{M}(R) we have Ass(N_{i}) = Z(σ[N_{i}]), i = 1, 2. Let S be a simple module such that S can be embedded in N_{1} and N_{2}, then we may define a non zero homomorphism f: N_{1} → N_{2}, so N_{1} is not a (R, End(_{R}M)^{opp})–subbimodule, which is a contradiction. Hence Ass(N_{1}) ∩ Ass(N_{2}) = ∅. The fourth condition in the definition of σ–decomposable modules is also satisfied. Therefore M = N_{1} ⊕ N_{2} is a σ–decomposition.

(b) ⇒ (a) We only need to prove that M = N_{1} ⊕ N_{2} as right End(_{R}M)– modules, or equivalently that Hom_{R}(N_{i}, N_{j}) = 0 if i ≠ j. Let, for instance, 0 ≠ f: N_{1} → N_{2}, then there exists a simple submodule S_{i} of N_{i}, i = 1, 2, and a non zero homomorphism g: E_{M}(S_{1}) → S_{2}, which is a contradiction. □

We now prove that also in this case the decomposition built in Proposition 12 is compatible with any decomposition of M in fully invariant submodules.

18 Corollary. Let M be a locally finite left R–module such that M is an injective cogenerator of σ[M] and let M = ⊕_{α}M_{α} the decomposition given in Proposition 12. Let M = N_{1} ⊕ N_{2} be a decomposition de M as (R, End(_{R}M))–bimodules, then for every index α ∈ Λ we have either M_{α} ⊆ N_{1} or M_{α} ⊆ N_{2}.

6. Decomposition of coalgebras and comodules.

19 Example. One application of the developed theory may be found in the study of the dual coalgebra of an algebra. It is well known that every locally finite left R–module has a uniquely determined structure of right R^{∘}–comodule, where R^{∘} is the dual coalgebra of R. Even more, the closed subcategory of all locally finite left R–modules can be identified with the subcategory σ[_{R}R^{∘}] of R –Mod. In this case R^{∘} is an injective cogenerator of σ[_{R}R^{∘}], so we have a decomposition of _{R}R^{∘} in indecomposable components. Using Proposition 12 this decomposition can be performed to any locally finite left R–module.

20 Example. Mutatis mutandi the same results may be obtained in the study of a coalgebra C. In this case the right C–comodules coincide with the rational left C^{∗}– modules and these are exactly the modules in σ[_{C^{∗}}C]. The decomposition of the coalgebra C we obtain in that way is exactly the decomposition given in [4]. But in addition we obtain a decomposition theory of comodules. To prove that both decomposition theories are the same we only need to show that if C is a coalgebra, then there exists a ring isomorphism between End(_{C^{∗}}C)^{op} and C^{∗}. So a decomposition of C as (C^{∗}, End(_{C^{∗}}C)^{opp})–bimodule is exactly a decomposition of C as (C^{∗}, C^{∗})–bimodule.

We have in this case a like primary decomposition of any locally finite left R–module. This is an extension of the cocommutative theory for comodules, as is developed, for instance in [5]. The main essential difference consists in considering sets of maximal cofinite ideals instead of single maximal ideals to define the primary components.

Decomposition of comodules has been also studied for instance in [8]. However, the decomposition theory in that paper is unsatisfactory as the next example shows.

21 Example. Let us consider R = (\begin{matrix}K & K \\ 0 & K \end{matrix} ) the ring of upper triangular matrices over a

field K and P = (\begin{matrix}K & K \\ 0 & 0 \end{matrix} ), Q = (\begin{matrix}0 & K \\ 0 & K \end{matrix} ). We have that P and Q are prime

ideals of R and 0 = QP ⊂ Q∩P = (\begin{matrix}0 & K \\ 0 & 0 \end{matrix} ). Then there exists a link Q ⇝ P.

Let us consider the left R–module X = (\begin{matrix}K & 0 \\ K & 0 \end{matrix}). We may build a composition

series

(\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}) ⊆ (\begin{matrix}K & 0 \\ 0 & 0 \end{matrix}) ⊆ (\begin{matrix}K & 0 \\ K & 0 \end{matrix})

In this case R^{∘} is a finite–dimensional coalgebra and R^{∘∗} ≅ R, so locally finite R–modules coincide with rational R–modules.

Let us call T = (\begin{matrix}K & 0 \\ 0 & 0 \end{matrix}). We have Ann^{l}(T) = Q and Ann^{l}(X/T) = P and

a non–splitting short exact sequence 0 → T → X → X/T → 0, so X is uniform.

Let us define M = X ⊕ (X/T), then Soc(M) = T ⊕ (X/T) and also Ass(M) = {P, Q}, Ass(X) = {Q} and Ass(X/T) = {P }. We have that M is indecomposable in the sense of Xu–Shum–Fong; it is also σ–indecomposable.

On the other hand, if we consider X, remember that Ass(X) = {Q}, then it satisfies the hypothesis in [8, Theorem 4.11], and following Xu–Shum– Fong M must be a direct sum of indecomposables. The decomposition in [8] is built from the simple submodules, but in this case the only simple submodule is T and we can not reach M unless using the maximal cofinite ideal P or equivalently the simple X/T, and this can not obtained as a simple submodule of X. So X can not be decomposed using [8, Theorem 4.15].

References

[1] J. S. Golan, Linear topologies on a ring: an overview, Pitman Research Notes in Math. Series, 159, Longman Scientific & Technical, 1987.

[2] K. R. Goodearl and R. B. Warfield, An introduction to noncommutative noetherian rings, London Math. Soc. Student Series, 16, Cambridge Univ. Press, 1989.

[3] P. Jara, A. Verschoren, and C. Vidal, Localization and sheaves: a relative point of view, Pitman Research Notes in Math. Series, 339, Longman, 1995.

[4] S. Montgomery, Indecomposable coalgebras, simple comodules and pointed Hopf algebras, Proc. Amer. Math. Soc. 123 (1995), 2343–2351.

[5] C. Nastasescu and B. Torrecillas, Torsion theories for coalgebras, J. Pure Appl. Algebra 97 (1994), 203–220.

[6] B. Stenström, Rings of quotients, Springer–Verlag, Berlin, 1975.

[7] R. Wisbauer, Foundation of module and ring theory. A handbook for study and research, Algebra, Logic and Applications Series, volume 3, Gordon and Breach Science Publ., 1991.

[8] Y. Xu, K. P. Shum, and Y. Fong, A decomposition theory of comodules, J. Algebra 170 (1994), 880–906.