Flat Semimodules

To my dearest friend Najla Ali We introduce and investigate flat semimodules and k-flat semimodules. We hope these concepts will have the same importance in semimodule theory as in the theory of rings and modules. 1. Introduction. We introduce the notion of flat and k-flat. In Section 2, we study the structure ensuing from these notions. Proposition 2.4 asserts that V is flat if and only if (V ⊗ R −) preserves the exactness of all right-regular short exact sequences. Proposition 2.5 gives necessary and sufficient conditions for a projective semimodule to be k-flat. In Section 3, Proposition 3.3 gives the relation between flatness and in-jectivity. In Section 4, Proposition 4.1 characterizes the k-flat cancellable semimodules with the left ideals. Proposition 4.4 describes the relationship between the notions of projectivity and flatness for a certain restricted class of semirings and semimodules. Throughout, R will denote a semiring with identity 1. All semimodules M will be left R-semimodules, except at cited places, and in all cases are unitary semimodules, that


Introduction.
We introduce the notion of flat and k-flat.In Section 2, we study the structure ensuing from these notions.Proposition 2.4 asserts that V is flat if and only if (V ⊗ R −) preserves the exactness of all right-regular short exact sequences.Proposition 2.5 gives necessary and sufficient conditions for a projective semimodule to be k-flat.In Section 3, Proposition 3.3 gives the relation between flatness and injectivity.In Section 4, Proposition 4.1 characterizes the k-flat cancellable semimodules with the left ideals.Proposition 4.4 describes the relationship between the notions of projectivity and flatness for a certain restricted class of semirings and semimodules.Throughout, R will denote a semiring with identity 1.All semimodules M will be left R-semimodules, except at cited places, and in all cases are unitary semimodules, that is, 1 • m = m for all m ∈ M (m • 1 = m for all m ∈ M) for all left R-semimodules R M (resp., for all right R-semimodule M R ).
We recall here (cf.[1,2,4,7,8]) the following facts.(c) A semiring R is called completely subtractive if R R is a completely subtractive semimodule; and a left R-semimodule M is called completely subtractive if and only if for every subsemimodule N of M, N is subtractive.
(d) A semimodule M is said to be free R-semimodule if M has a basis over R. (e) A semimodule C is said to be semicogenerated by U when there is a homomorphism ϕ : M → Π A C such that ker θ = 0.A semimodule C is said to be a semicogenerator when C semicogenerates every left R-semimodule M.
(f) Let α : M → N be a homomorphism of semimodules.The subsemimodule Im α of N is defined as follows: Im α = {n ∈ N : n+α(m ) = α(m) for some m, m ∈ M}.Also α is said to be a semimonomorphism if ker α = 0, to be a semi-isomorphism if α is surjective and Ker α = 0, to be an isomorphism if α is injective and surjective, to be i-regular if α(M) = Im α, to be k-regular if for a, a ∈ A, α(a) = α(a ) implying a + k = a + k for some k, k ∈ ker α, and to be regular if it is both i-regular and k-regular.
(g) An R-semimodule M is said to be k-regular if there exist a free R-semimodule F and a surjective R-homomorphism α : semimodules} is a semigroup under addition.If M, N, and U are R-semimodules and α : M → N is a homomorphism, then Hom(α, I U ) : Hom R (N, U) → Hom R (M, U ) is given by Hom(α, I U )γ = γα, where I U is the identity on U.
(k) If M is a right R-semimodule, N is a left R-semimodule, and T is an N-semimodule, then a function θ : M × N → T is R-balanced if and only if, for all m, m ∈ M, for all n, n ∈ N, and for all r ∈ R, we have (1) Let R be a semiring, let M be a right R-semimodule, and let N be a left R-semimodule.Let A be the set M ×N, and let U be the N-semimodule ⊕ A N×⊕ A N. Let W be the subset of U consisting of all elements of the following forms: ( ), for m and m in M, n and n in N, and r in R, and where α[m, n] is the function from M × N to N which sends (m, n) to 1 and sends every other element of M × N to 0. Let U be the N-subsemimodule of U generated by W . Define N congruence relation ≡ on ⊕ A N by setting α ≡ α if and only if there exists an element (β, γ) ∈ U such that α + β = α + γ.The factor N-semimodule ⊕ A N/ ≡ will be denoted by M ⊗ R N, and is called the tensor product of M and N over R.
(A) A left R-semimodule P is said to be projective semimodule if and only if for each surjective R-homomorphism ϕ : M → N, the induced homomorphism ϕ : Hom R (P , M) → Hom R (P , N) is surjective.

Flat and k-flat semimodules.
In this section, we discuss the structure of flat and kflat semimodules.Proposition 2.4 asserts that V is flat if and only if (V ⊗ R −) preserves the exactness of all left k-regular right regular short sequences.In Proposition 2.5, we give the necessary and sufficient condition for the projective right semimodule to be k-flat relative to a cancellable left semimodule.
Our next result shows that the class of flat and k-flat semimodules is closed under direct sums.
Proof.Let M be a left R-semimodule and K a subsemimodule of M. Consider the following commutative diagram: where (2.2) Consider the following exact sequence: Our next result gives a necessary and sufficient condition for a projective semimodule to be k-flat relative to a cancellable semimodule M. Proposition 2.5.Let V R be projective and R M cancellable.Then, V is Mk-flat if and only if the functor (V ⊗ R −) preserves the exactness of all left k-regular right regular short exact sequences (2.5) Consider the following exact sequence: Since V is projective and M is cancellable, then by using [9, Proposition 1.16], 3. Flatness via injectivity.We will discuss the relation between the injectivity and flatness.By (•) * we mean the functor Hom N (−,C), where C is a fixed injective semicogenerator cancellative N-semimodule.
We state and prove the following lemma, analogous to the one on modules which is needed in the proof of Proposition 3.3.Lemma 3.2.Let R be a semiring, let M and M be left R-semimodules, and let U be a right R-semimodule.Let T be a cancellative N-semimodule.If α : M → M is an R-homomorphism, then there exist N-isomorphisms ϕ and ϕ such that the following diagram commutes: Proof.By [7, Proposition 14.15], there exists an N-isomorphism given by ϕ(γ) : u ⊗ m γ(m)u.Then with a parallel definition for ϕ , we have and the diagram commutes.
Proof.(1) Let K be a subsemimodule of M. Since V is Mk-flat, then the sequence 0 → V ⊗K where ϕ and ϕ are N-isomorphisms.It follows that the top row is proper exact if and only if the bottom row is proper exact, whence by [6, Proposition 3.1], V * is injective. ( is proper exact.Again by the above diagram, is proper exact.Hence, the sequence is exact.Since C is a semicogenerator, then by [3,Proposition 4.1], the sequence 0 4. Cancellable semimodules.In this section, we deal with cancellable semimodules.We characterize k-flat cancellable semimodules by means of left ideals.
Proposition 4.1.The following statements about a cancellable right R-semimodule V are equivalent: (1) V is k-flat relative to R R; (2) for each (finitely generated) left ideal Proof.(1)⇒(2).Since V is cancellable, then by using [7, Proposition 14.16], V ⊗ R R V .Consider the following commutative diagram: where θ is the isomorphism of [7,Proposition 14.16].Since ψ : V × I → V I given by ψ(v, i) = vi is an R-balanced function, then by using [7, Proposition 14.14], there is an exact unique N-homomorphism ϕ : (2)⇒(1).Again consider the above diagram.Let I be any left ideal of R and let whence Σv i a i = Σv i a i .Now consider the following diagram, where i K : K → I is the inclusion map: Proof.Let i K : K → M be the inclusion homomorphism.By [7,Proposition 14.16], R ⊗ R K K and R ⊗ R M M. Consider the following commutative diagram: Proof.The proof is immediate from Propositions 2.3 and 4.2.
In module theory every projective module is flat.Now we see that this is true for certain special semimodules.Proof.We only need to show that M is cancellable.Since M is k-regular, then there exists a free R-semimodule F such that ϕ : Proof.By using Corollary 4.3, V is Mk-flat.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: (a) A semiring R is said to satisfy the left cancellation law if and only if for all a, b, c ∈ R, a + b = a + c ⇒ b = c.A semimodule M is said to satisfy the left cancellation law if for all m, m ,m ∈ M, m + m = m + m ⇒ m = m .(b) We say that a nonempty subset N of a left semimodule M is subtractive if and only if for all m, m ∈ M, m, m + m in N imply m in N.

Proposition 2 . 4 .
Let M be a left R-semimodule.A right R-semimodule V is M flat if and only if the functor (V ⊗ R −) preserves the exactness of all left k-regular right regular short exact sequences with middle term M:

Proposition 4 . 4 .Corollary 4 . 5 .
Let M be a cancellable left R-semimodule, where R is a cancellative completely subtractive semiring.Then every k-regular projective R-semimodule P is Mkflat.Proof.By using [5, Theorem 19], P is isomorphic to a direct summand of a free semimodule F .By Corollary 4.3, F is Mk-flat.Hence, by using Proposition 2.3, P is Mkflat.Let M be a k-regular left R-semimodule and R a cancellative completely subtractive semiring.Then every k-regular projective R-semimodule P is Mk-flat.