BLOCK DECOMPOSITION FOR MODULES

Block decomposition for rings has been introduced and shown to be unique in the literature (see [T. Y. Lam, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991]). Applying annihilator submodules, we extend this definition to modules and show that every module M has a unique block decomposition M = ⊕n i=1 Mi where each Mi is an annihilator submodule. We also show that the block decomposition for any ring R and the block decomposition for the module RR, are identical. Block decomposition provides us with a decomposition for End(M) because End(M) ∼= ∏n i=1 End(Mi). Mathematics Subject Classification (2010): 16D70, 16D50, 16D80


Introduction
Any ring with unit satisfying some finiteness conditions on ideals, is isomorphic to the direct product of a finite family of indecomposable rings and this representation is unique [4, 2.22], in the following sense. Let each R i and each S j be an indecomposable ring. If there is an isomorphism ∆ : n i=1 R i −→ k j=1 S j , then k = n, and, after a re indexing, there is isomorphisms ∆ i : R i −→ S i for 1 ≤ i ≤ n such that ∆ = n i=1 ∆ i . But the decomposition structure for modules is not unique. Even if decomposition structure is unique for certain modules [4, 19.22, Krull-Schmidt Theorem], it is not unique in the above sense. The reason is that viewing a ring as a module on itself, the class of submodules is not similar to the class of ideals.
Applying annihilator submodules and adjusted techniques, we will be able to introduce a ring type decomposition for modules. Also, with the same techniques, we show that any left or right faithful ring, not necessary having unit and satisfying even weaker finiteness conditions in [4, 2.22], is decomposable.
In Section 1, we outline a general theory in additive group format so that a large part of the proofs of some theorems would become special case of a theory that is useful in this context and is of intrinsic interest by itself. In Section 2, 188 H. KHABAZIAN we introduce annihilator submodules, then standard facts are collected, some basic properties are developed and we apply Proposition 2.16 to derive the existence of the block decomposition for rings and modules. In Section 3, we show that the block decomposition for a ring R and block decomposition for R R are identical.
In this paper, for any set S of subgroups of an additive group, we set Σ(S) = I∈S I and S is said to be independent if I∈S I is a direct sum. Also, we use the notation (S) instead of Σ(S) to indicate that S is independent.
For any class C of subgroups, a C-subgroup means a subgroup from the class C, the class of minimal C-subgroups is denoted by C mn and the class of maximal C-subgroups is denoted by C mx .
For classes C and F of subgroups, the class of subgroups which are a C-subgroup and an F-subgroup is denoted by C ∩ F.
For an additive group M , the set of C-subgroups of M is denoted by C : M , For an additive group M and K ⊆ M , the set of C-subgroups of M containing K is denoted by C ⊇ K , the set of C-subgroups of M contained in K is denoted by C ⊆ K , and the set of C-subgroups of M not contained in K is denoted by For any additive groups U , V and W , any X ⊆ U , Z ⊆ W and multiplication and we say that X is V -faithful if ann V (X) = 0. For the case V × V −→ W , we set (Z : X) r = {v ∈ V | Xv ⊆ Z}, ann r (X) = {v ∈ V | Xv = 0} and we say X is right faithful if ann r (X) = 0.

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For any family S of subsets of a set, we set Int(S) = I∈S I and Un(S) = I∈S I.
In the context, C and G designate arbitrary classes of subgroups.

Preliminaries
Definition 2.1. Let M be an additive group and K be a subgroup.
(2) We say that K is C-indecomposable if for any C-subgroups I and J, K = I ⊕ J implies I = 0 or J = 0.
(3) We say that K is C-uniform if K = 0 and for any C-subgroups I, J ⊆ K, Also, the class of C-summand subgroups is denoted by C ⊕ . Recall that according to the above notifications, C ∩ C ⊕ is the class of C-summand C-subgroups, (C ∩ C ⊕ ) mn is the class of minimal C-summand C-subgroups, and Σ (C ∩ C ⊕ ) mn : M is the sum of the minimal C-summand C-subgroups of M . (2) For any G-subgroups J and K, K ∩ J = 0 implies J ⊆ K .
Also, M is called G-organized if it has a G-organizer map.
If is a G-organizer map, then for any G-subgroups J and K, K ⊆ J implies J ⊆ K because we have K ∩ J = 0. Lemma 2.3. Let M be an additive group and K be a nonzero subgroup. If every independent set of C-subgroups contained in K is finite, then K contains a C-uniform C-subgroup.
Proof. Temporarily suppose that it is not so. There exist nonzero C-subgroups is an infinite independent set of nonzero C-subgroups contained in K which is a contradiction. (1) For any G-subgroup K, K ∩ K = 0.
(2) For any G-subgroups J and K, K ∩ J = 0 implies J ⊆ K .

H. KHABAZIAN
Then, for every G-subgroup K we have Σ G mn ⊆ K = K , Σ G mn ⊆ K = K, M = K ⊕ K and K = K. Also is a G-organizer map.
Lemma 2.6. Let M be a G-intersection (generalized G-intersection) additive group and be a G-organizer map. For any finite set (any set) A of G-subgroups, (2) For each J ∈ A, L ∩ J = 0, implying J ⊆ L . Thus, A ⊆ G mn ⊆ L . Also for each I ∈ A and J ∈ G mn : M with J ∈ A, I ∩ J = 0, implying J ⊆ I . So, Lemma 2.7. Let M be a G-intersection additive group and be a G-organizer map.
If I is a G-summand G-subgroup, then for any G-subgroup J we have   (4) As we showed in proof of (1), Lemma 2.8. Let M be a G-intersection and G-organized additive group.
(1) G mn : M is an independent set.   (1) Every G-subgroup K is G-summand and M = K ⊕ K .
(2) We have M = 0, so 0 is a G-subgroup. Applying (1) completes the proof .     (7) Every G-subgroup is the intersection of a finite set of maximal G-subgroups.
In this case, M is generalized G-intersection, and if every G-subgroup is a Gsummand, then M is G-semisimple.
. K n is a G-subgroups. Also, J n+1 ∩ K n = 0 and J n+1 ⊆ K n+1 . Thus K n ⊂ K n+1 , which is a contradiction.  Lemma 2.14. Let M be a G-intersection additive group and be a G-organizer map. Consider P as the class of G-summand G-subgroups. Then is a P-organizer map.
Proof. Let K be a P-subgroup. It follows from Lemma 2.5 that K is also a P-subgroup.  (2) M is P-intersection.
(2) Every G-subgroup is an intersection of maximal G-subgroups.

Proof. (1) It is enough to show that the sum of any set A of minimal G-subgroups
is a G-subgroups by Lemma 2.9. Set L = Int{I | I ∈ A}. A = G mn ⊆ L by Lemma 2.6, implying Σ(A) = L by Lemma 2.9.
(2) Let K be an G-subgroup. Set A = G mn ⊆ K . Then, K = Σ(A) by Lemma 2.9, so K = Int{I | I ∈ A} by Lemma 2.6 and (1). On the other hand, I is a maximal G-subgroup for each I ∈ A by Lemma 2.5.  (1) For every P ⊆ M , ann M (ann R (M/P )) is denoted by P • .
(2) P ⊆ M is called an annihilator if ann M (ann R (P )) = P .
(3) M is said to be cofaithful if R is M -faithful. It should be noted that the use of the term "cofaithful" hear is not the same as the usage in [1].
(4) M is said to be block indecomposable if for any block orthogonal R-modules N and P , M ∼ = N × P implies either N = 0 or P = 0.
(5) If there exist pairwise block orthogonal block indecomposable modules M 1 , Also, in the category of modules, the class of annihilators is denoted by A, the class of submodules is denoted by M, the class of annihilator submodules is denoted by AM, and the class of submodules P for which P ∩ P • = 0 and P •• = P is denoted by C.
It is obvious that for any P ⊆ M , P • is an annihilator submodule, so P •• is also an annihilator submodule.     (1) Any two annihilator submodules with zero intersection are block orthogonal.
(2) For any block orthogonal submodules K and J with K ∩ J = 0, if K ⊕ J is an annihilator, then K and J are annihilators.
(3) Any submodule which is a block indecomposable module, is AM-indecomposable.
Proof. The first statement follows from Lemma 3.4.  for each 1 ≤ j ≤ n. I j is N -faithful for each 1 ≤ j ≤ n, so I 1 I 2 · · · I n is N -faithful, thus ∩ n i=1 I i is N -faithful. Therefore ann N (ann R (M )) = 0.
Proposition 3.7. Let R be a ring and M be an R-module. Every AM-summand annihilator submodule K is a C-summand C-submodule, M = K ⊕ K • and K and K • are block orthogonal modules. Also the following conditions are equivalent.
(1) K is a block indecomposable module.
Proof. There exists an annihilator submodule J with M = K ⊕ J. Now applying Lemma 3.5 completes the proof.
(2) If M j is a block indecomposable module for some 1 ≤ j ≤ n, then ι j (M j ) is a minimal C-summand C-submodule.
Notice that according to Lemma 4.2, in the category of left faithful rings, the classes I ∩ I ⊕ , lAI ∩ lAI ⊕ and lC ∩ lC ⊕ are identical.  Proof. The same as the proof of Theorem 3.12.
The following theorem gives us the uniqueness of the the block decomposition.
If ∆ : n i=1 R i −→ k j=1 S j is an isomorphism, then k = n, and after a re indexing, there exist isomorphisms ∆ i : R i −→ S i for 1 ≤ i ≤ n such that ∆ = n i=1 ∆ i .
Proof. The same as the proof of Theorem 3.13. Theorem 4.6. A left faithful ring R is decomposable if and only if R R is block decomposable. In this case, the block decomposition for R and the block decomposition for R R are identical.
Proof. We have lAI ∩ lAI ⊕ : R = AM ∩ AM ⊕ : R R . Applying Theorem 3.11 and Theorem 4.3 completes the proof.