© Hindawi Publishing Corp. ON SOME PROPERTIES OF ⊕-SUPPLEMENTED MODULES

Am oduleM is ⊕-supplemented if every submodule of M has a supplement which is a direct summand of M. In this paper, we show that a quotient of a ⊕-supplemented module is not in general ⊕-supplemented. We prove that over a commutative ring R, every finitely generated ⊕-supplemented R-module M having dual Goldie dimension less than or equal to three is a direct sum of local modules. It is also shown that a ring R is semisimple if and only if the class of ⊕-supplemented R-modules coincides with the class of injective R-modules. The structure of ⊕-supplemented modules over a commutative principal ideal ring is completely determined.


Introduction.
All rings considered in this paper will be associative with an identity element.Unless otherwise mentioned, all modules will be left unitary modules.Let R be a ring and M an R-module.Let A and P be submodules of M. The submodule P is called a supplement of A if it is minimal with respect to the property A + P = M. Any L ≤ M which is the supplement of an N ≤ M will be called a supplement submodule of M. If every submodule U of M has a supplement in M, we call M complemented.In [25, page 331], Zöschinger shows that over a discrete valuation ring R, every complemented R-module satisfies the following property (P ): every submodule has a supplement which is a direct summand.He also remarked in [25, page 333] that every module of the form M (R/a 1 ) × ••• × (R/a n ), where R is a commutative local ring and a i (1 ≤ i ≤ n) are ideals of R, satisfies (P ).In [12, page 95], Mohamed and Müller called a module ⊕-supplemented if it satisfies property (P ).
On the other hand, let U and V be submodules of a module M. The submodule V is called a complement of U in M if V is maximal with respect to the property V ∩U = 0.In [17] Smith and Tercan investigate the following property which they called (C 11 ): every submodule of M has a complement which is a direct summand of M. So, it was natural to introduce a dual notion of (C 11 ) which we called (D 11 ) (see [6,7]).It turns out that modules satisfying (D 11 ) are exactly the ⊕-supplemented modules.A module M is called a completely ⊕-supplemented (see [5]) (or satisfies (D +  11 ) in our terminology, see [6,7]) if every direct summand of M is ⊕-supplemented.
Our paper is divided into four sections.The purpose of Section 2 is to answer the following natural question: is any factor module of a ⊕-supplemented module ⊕-supplemented?Some relevant counterexamples are given.
In Section 3 we prove that, over a commutative ring, every finitely generated ⊕-supplemented module having dual Goldie dimension less than or equal to three is a direct sum of local modules.
Section 4 describes the structure of ⊕-supplemented modules over commutative principal ideal rings.
In the last section we determine the class of rings R with the property that every ⊕-supplemented R-module is injective.These turn out to be the class of all left Noetherian V -rings (Proposition 5.3).It is also shown that a ring R is semisimple if and only if the class of ⊕-supplemented R-modules coincides with the class of injective R-modules (Proposition 5.5).
For an arbitrary module M, we will denote by Rad(M) the Jacobson radical of M. The injective hull of M will be denoted by E(M).The annihilator of M will be denoted by Ann R (M).A submodule A nonzero module H is called hollow if every proper submodule is small in H and is called local if the sum of all its proper submodules is also a proper submodule.We notice that a local module is just a cyclic hollow module.

Quotients of ⊕-supplemented modules. By [23, corollary on page 45],
every factor module of a complemented module is complemented.Now, let M be a ⊕-supplemented module.In this section we will answer the following natural question: is any factor module of M ⊕-supplemented?
First, we mention the following result, which we will use frequently in the sequel.
Proposition 2.1 [6,Proposition 1].The following are equivalent for a module M: A commutative ring R is a valuation ring if it satisfies one of the following three equivalent conditions: (i) for any two elements a and b, either a divides b or b divides a; (ii) the ideals of R are linearly ordered by inclusion; (iii) R is a local ring and every finitely generated ideal is principal.A module M is called finitely presented if M F/K for some finitely generated free module F and finitely generated submodule K of F .An important result about these modules is that if M is finitely presented and M F/G, where F is a finitely generated free module, then G is also finitely generated (see [2]).
Example 2.2.Let R be a commutative local ring which is not a valuation ring and let n ≥ 2. By [21,Theorem 2], there exists a finitely presented indecomposable module M = R (n) /K which cannot be generated by fewer than n elements.By [6, Corollary 1], R (n) is ⊕-supplemented.However M is not ⊕-supplemented [6, Proposition 2].
The dual Goldie dimension of an R-module, denoted by corank( R M), was introduced by Varadarajan in [19].If M = 0, the corank of M is defined as 0. Let M ≠ 0 and k an integer greater than or equal to one.If there is an epimor- we say that the corank( R M) = ∞.It was shown in [14,19] that the corank( R M) < ∞ if and only if there is an epimorphism f : M → k i=1 H i , where H i is hollow and ker(f ) is small in M.
As in [20], a module M has the exchange property if for any module G, where with M M, there are submodules Before proceeding any further, we consider another example (note that the module considered is decomposable).
Example 2.3.Let R be a commutative local ring which is not a valuation ring.Let a and b be elements of R, neither of them divides the other.By taking a suitable quotient ring, we may assume (a)∩(b) = 0 and am = bm = 0, where m is the maximal ideal of R. Let F be a free module with generators x 1 , x 2 , and x 3 .Let K be the submodule generated by ax 1 − bx 2 and let M = F/K.Thus, which gives bβ = aα and then a = bβα , which is a contradiction.Since These examples show that a factor module of a ⊕-supplemented module is not in general ⊕-supplemented.
Proposition 2.5 deals with a special case of factor modules of ⊕-supplemented modules.First we prove the following lemma.
Lemma 2.4.Let M be a nonzero module and let U be a submodule of Proposition 2.5.Let M be a nonzero module and let U be a submodule of (2.4) Hence, Corollary 2.6.Let M be an R-module and P (M) the sum of all its radical submodules.
Proof.By Proposition 2.5, it suffices to prove that f (P(M)) ≤ P (M) for each f ∈ End R (M).Let N be a radical submodule of M and let f be an endomorphism of M and g its restriction to N. By [1, Proposition 9.14], g( Thus, Rad(f (N)) = f (N).This implies that f (N) ≤ P (M), and the corollary is proved.
We recall that a module M is called semi-Artinian if every nonzero quotient module of M has nonzero socle.For a module R M, we define ( By [18, Chapter VIII, Section 2, Corollary 2.2], if R is a left Noetherian ring and R M a semi-Artinian left R-module, then M is the sum of its submodules of finite length. If R is a commutative Noetherian ring and M is an R-module, then Sa(M) = L(M), the sum of all Artinian submodules of M.
Proof.By Proposition 2.5, it suffices to prove that f (Sa(M)) ≤ Sa(M) for each f ∈ End R (M).Let U be a semi-Artinian submodule of M and let f be an endomorphism of M and g its restriction to U. Thus U/Ker(g) g(U).Hence f (U) U/Ker(g).But it is easy to check that U/Ker(g) is a semi-Artinian module.Therefore, f (U) is semi-Artinian.

Some properties of finitely generated ⊕-supplemented modules. A module M is called supplemented if for any two submodules A and B with
The proof of the next result is taken from [6, Lemma 2], but is given for the sake of completeness.Lemma 3.1.Let M be a ⊕-supplemented R-module.If M contains a maximal submodule, then M contains a local direct summand.
Proof.Let L be a maximal submodule of M. Since M is ⊕-supplemented, there exists a direct summand K of M such that K is a supplement of L in M. Then for any proper submodule X of K, X is contained in L since L is a maximal submodule and L + X is a proper submodule of M by minimality of K. Hence X ≤ L ∩ K and X is small in K by [12,Lemma 4.5].Thus K is a hollow module, and the lemma is proved.Proposition 3.2.If M is a ⊕-supplemented module such that Rad(M) is small in M, then M can be written as an irredundant sum of local direct summands of M.
Proof.Since Rad(M) is small in M, M contains a maximal submodule and hence M contains a local direct summand by Lemma 3.1.Let N be the sum of all local direct summands of M. If N is a proper submodule of M, then there exists a maximal submodule L of M such that N ≤ L (see [8, Proposition 9 and Theorem 8]).Let P be a direct summand of M such that P is a supplement of L in M. Note that P is a local module (see the proof of Lemma 3.1) and hence it is contained in N, so M = L + P ≤ L + N = L.This is a contradiction.Hence we have N = M. Now let M = i∈I L i where each L i is a local direct summand of M.Then, and each for some subset K ⊆ I. Thus M = k∈K L k since Rad(M) is small in M. Clearly, the sum k∈K L k is irredundant.

Corollary 3.3. Let R be a commutative ring and M a finitely generated
(ii) In the same example, we have that Then K is not an irredundant sum of local direct summands.This example shows that, in general, a direct summand of a module which is written as an irredundant sum of local direct summands does not have the same property.
Proposition 3.5.Let M be a finitely generated ⊕-supplemented module such that k = corank(M) ≤ 2. Then M is a direct sum of local modules.
Proof.It is clear that if k = 1, then M is a local module.Now suppose that k = 2. Since M is ⊕-supplemented, M contains a local direct summand H (Lemma 3.1).Let K be a submodule of M such that M = H ⊕K. By [14, Corollary 1.9], we have corank(K) = 1 and hence K is a local module (see [19,Proposition 1.11]).Thus M is a direct sum of local modules, as required.
Our next objective is to prove that over a commutative ring, if M is a finitely generated ⊕-supplemented module with corank(M) = 3, then M is a direct sum of local modules.We first prove the following generalization of [11,Lemma 2.3].
Proof.We use induction over n.Assume that Proposition 1], L n has the exchange property.Thus, M = L n ⊕ L ⊕ N for some submodules L and N of M with L ≤ L and N ≤ N. Let N and L be two submodules of , and the lemma is proved.
Corollary 3.7.Suppose that R is commutative or left Noetherian.Let L 1 , L 2 ,...,L n be hollow local direct summands of a module M. If L i L j for all i ≠ j, then n i=1 L i is direct and is a direct summand of M.
Proof.This is a consequence of [4, Theorems 4.1 and 4.2] and Lemma 3.6.Proposition 3.8.Suppose that R is a commutative ring.Let M be a finitely generated ⊕-supplemented module such that all the hollow direct summands of M are isomorphic.Then M is a direct sum of hollow local modules.

Proof. By Proposition 3.2, we can write
( Ann Therefore all hollow local direct summands of M are isomorphic to R/I, where I = Ann R (M).Let H be a local submodule of M such that H is not small in M.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning is a local ring by [4, Theorem 4.1].Since Rx 3 has the exchange property [20, Proposition 1], there are submodules H ≤ H and N

Corollary 3 . 9 .Corollary 4 . 3 .
and H ∩ N is small in N (Proposition 2.1).It follows that N M/N H/(H ∩ N).Hence, N is a local module.This implies that Ann R (N ) = I and Ann R (H/(H ∩ N)) = I.Thus, the set {r ∈ R | r x ∈ N} = I, where H = Rx.Let y ∈ H ∩ N.There exists α ∈ R with y = αx.So α ∈ I and hence y = 0 since I ⊆ Ann R (H).Therefore H ∩N = 0 and M = H ⊕N. It follows that every nonsmall local submodule of M is a direct summand of M. Note that corank(M) < ∞ (Corollary 3.3).Applying [23, corollary on page 45] and [8, Proposition 9], we get that M is a direct sum of local modules.Let R be a commutative ring and M a finitely generated ⊕-supplemented module with corank(M) = 3.Then M is a direct sum of local modules.Proof.Let F 0 be an irredundant set of representatives of the local direct summands of M (F 0 is not empty by Lemma 3.1).By Corollary 3.7, Card(F 0 ) ≤ 3.If Card(F 0 ) = 3, then M is a direct sum of local modules (Corollary 3.7).If Card(F 0 ) = 2 and F 0 = {L 1 ,L 2 }, then there exists a submodule L 3 of M such that (i) M is ⊕-supplemented;(ii) M = K(M) and K m (M) is ⊕-supplemented for all m ∈ Ω.Let R be a commutative principal ideal ring (not necessarily a domain) and M an R-module.The following conditions are equivalent:

Corollary 4 . 5 .
is the quotient field of R m and B m (1,...,n m ) denotes the direct sum of arbitrarily many copies of R m /mR m ,...,R m /(mR m ) nm , for some positive integer n m .Proof.See Proposition 4.2, [13, Proposition 2.2(b)], and Theorem 4.1.Proposition 4.4 (see [7, Corollary 2.2]).Let R be a commutative Noetherian ring and M an R-module.The following assertions are equivalent: (i) M is completely ⊕-supplemented; (ii) M = K(M) and K m (M) is completely ⊕-supplemented for all m ∈ Ω.Let R be a commutative principal ideal ring (not necessarily a domain) and M an R-module.Then M is ⊕-supplemented if and only if M is completely ⊕-supplemented.Proof.By Proposition 4.4 and the proof of Theorem 4.1, it suffices to prove the result for an R-module M over a local principal ideal domain R with maximal ideal m ≠ 0. If M is ⊕-supplemented, then M R a ⊕Q b ⊕(Q/R) c ⊕B(1,..., n), where Q is the quotient field of R and B(1,...,n) denotes the direct sum of arbitrarily many copies of R/m,...,R/m n (Theorem 4.1).By [7, Theorem 2.1], Q By Proposition 3.2, M = H 1 +H 2 +•••+H n , where each H i is a local direct summand of M and the sum n i=1 H i is irredundant.By [16, Corollary 4.6], M is supplemented.Therefore n = corank(M) by [14, Proposition 1.7] and [19, Lemma 2.36 and Theorem 2.39].The module M = (Rx 1 + Rx 2 ) ⊕ Rx 3 in Example 2.3 is not ⊕-supplemented.On the other hand, M can be written as follows: M where each H i is a local direct summand of M and n = corank(M).Proof.
module is injective.The ring R is called an SSI-ring if every semisimple left R-module is injective.Let M be a module with Rad(M) = 0. Then M is ⊕-supplemented if and only if M is semisimple.
Corollary 5.2.Let R be a left V -ring and M an R-module.Then M is ⊕supplemented if and only if M is semisimple.

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