CHARACTERIZATIONS OF PROJECTIVE AND k-PROJECTIVE SEMIMODULES

This paper deals with projective and k-projective semimodules. The results for projective semimodules are generalization of corresponding results for projective modules.


Introduction.
Throughout this paper, R denotes a semiring with identity 1, all semimodules M are left R-semimodules and in all cases are unitary semimodules, that is, 1 · m = m for all m ∈ M all left R-semimodule R M.
We recall here (cf. [1,2,3,4,5]) the following facts: is surjective; (e) a left R-semimodule P is Mk-projective if and only if it is projective with respect to every surjective k-regular homomorphism ϕ : M → N. In Section 2, we study the structure of k-projective semimodules. Proposition 2.2 shows that for a semimodule P , the class of all semimodules M such that P is Mkprojective is closed under subtractive subsemimodules, factor semimodules, and gives a sufficient condition for the class to be closed undertaking homomorphic images. Example 2.3 sheds light upon one difference between the structure of projectivity in module theory and semimodule theory. In Section 3, we characterize projective and k-projective semimodules via the Hom functor. Theorems 3.5 and 3.7 assert that P is M-projective (Mk-projective) if and only if Hom R (P , −) preserves the exactness of all proper exact sequences M α → M β → M , with β k-regular (both α and β k-regular).
2. k-projective semimodules. We study the structure of k-projective semimodules via the Hom function. We show that the class of all semimodules M, such that P is Mk-projective, is closed under subtractive subsemimodules, factor semimodule and undertaking homomorphic image for a k-regular homomorphism.
For proving Proposition 2.2 we need the following proposition, which is modified from [5, Theorem 2.6].
Proposition 2.1. Let R be a semiring, is a proper exact sequence of Abelian semigroups andᾱ is regular, is a proper exact sequence of Abelian semigroups andβ is regular, whereβ(ξ) = βξ. is exact withᾱ being regular. This means that the sequence is proper exact. (ii) can be proved by the same argument.
is a proper exact sequence with θ being regular, η being k-regular, and P is Mk-projective, then P is k-projective relative to both M and M .
Proof. Let Ψ : M → N be surjective k-regular homomorphism and α : P → N be homomorphism. Since η is surjective k-regular, then Ψ η is k-regular. Since P is Mk-projective, then there exists a homomorphism ϕ : P → M such that the following diagram commutative: Therefore P is M k-projective.
To prove that P is M k-projective. Let Ψ : M → N be a surjective k-regular homomorphism and set K = Ker Ψ . Since Ψ is surjective k-regular homomorphism, then M /K N. Defineθ : M /K → M/θ(K) by the ruleθ(m /K) = θ(m )/θ(K), andη : M/θ(K) → M by the ruleη(m/θ(K)) = η(m). Clearly, bothθ andη are well defined Clearlyη is surjective, andθ is injective. Since θ is i-regular, thenθ is i-regular. Now consider the following commutative diagram: Applying Hom R (P , −) to this diagram we have the commutative diagram Using Proposition 2.1, and since P is Mk-projective, then all rows and columns are proper exact sequence. We should show that (π K ) * is surjective. Let α ∈ Hom(P , M /K).
Let Ω(P ) be the collection of all semimodules M such that P is Mk-projective. The above results show that this class is closed under subtractive subsemimodules and give us a sufficient condition to be closed undertaking a homomorphic image. Since for every subsemimodule K of M, the canonical surjection π K : M → M/K is k-regular surjective, then the class Ω(P ) is closed under factor semimodules.
We know that in module theory any projective module is a direct summand of a free module. However, for arbitary semirings this is not true.
Define operations ⊕ and ⊗ on R by setting Clearly R is semiring. Let I + (R ) be the set of all additively idempotent elements of Since R is projective, as a left semimodule over itself, by [5,Corollary 15.13] we see that  (k, (0, {0})) has an additive inverse. Thus, ϕ(k, (0, {0})) also has an additive inverse. Now, every element of F is of the form (u α ), u α ∈ R . Since every nonzero element of R has no additive inverse, then every nonzero element of F has no additive inverse. Thus we have a contradiction. Therefore, I + (R ) is not a direct summand of a free R -semimodule.

Characterizations of projective and k-projective semimodules.
We characterize projective and k-projective semimodules via the Hom functor.
We state and prove the following lemma and corollaries which are needed in the proof of Theorem 3.5.

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 • Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation