ON INJECTIVE L-MODULES

The concepts of free modules, projective modules, injective modules,
and the like form an important area in module theory. The notion of free fuzzy modules was introduced by Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameri introduced the concept of projective and injective L -modules. In this paper, we give an alternate definition for injective L -modules and prove that a direct sum of L -modules is injective if and only if each L -module in the sum is injective. Also we prove that if J is an injective module and μ is an injective L -submodule of J , and if 0 → μ → f v → g η → 0 is a short exact sequence of L -modules, then ν ≃ μ ⊕ η .


Introduction
Though the notion of a fuzzy set was introduced by L. A. Zadeh in 1965, its application to algebraic concepts started only in 1971 when A. Rosenfeld introduced fuzzy subgroups of a group.Tremendous and rapid growth of fuzzy algebraic concepts resulted in a vast literature.The book of Mordeson and Malik [7] gives an account of all these up to 1998.The notion of L-modules as an extension of classical module theory is available in this book.However, there are many concepts in abstract algebra which are to be analyzed in the fuzzy context.The notion of free fuzzy modules was introduced by Muganda [8] as an extension of free modules in the fuzzy context.The concept of a free L-module is available in [7].Zahedi and Ameri [9] introduced the concepts of fuzzy projective and injective modules.
In our earlier paper [5], we introduced an alternate definition for a projective Lmodule and proved some related results.In this paper, in Section 2, we give the essential preliminaries and in Section 3, we give an alternate definition for an injective L-module and prove some results using this definition.Throughout this paper, unless otherwise stated, L(∨,∧,1,0) represents a complete Brouwerian lattice with maximal element "1" and minimal element "0;" R a ring with unity "1" and M a left module over R. "∨" denotes the supremum and "∧" the infimum in L. We call L a regular lattice if a ∧ b > 0 for all a,b > 0 in L. "⊆" denotes the inclusion and "⊂" the proper inclusion.The set of all L-subsets of M, that is, the set of all functions from M to L, is denoted by L M .For x ∈ M, a ∈ L, the L-subset which takes the value a at x and 0 elsewhere is denoted by a {x} .That is, (1.1)

Preliminaries
In this section, we review some definitions and results which will be used later.For details, reference may be made to Mordeson and Malik [7], for preliminaries regarding lattices Birkhoff [1], and for theory of modules, Goodearl and Warfield [2] and Hungerford [3].
For x ∈ M, Also for an arbitrary family where in the expression x = i∈I x i , at most finitely many x i 's are not equal to 0.
Definition 2.2 (see [7]).For µ ∈ L M , define the following: (i) µ * = {x ∈ M : µ(x) > 0}, called the support of µ, (ii) for a ∈ L, µ a = {x ∈ M : µ(x) ≥ a}, called the a-cut or a-level subset of µ, and µ > a = {x ∈ M : µ(x) > a}, called the strict a-cut or strict a-level subset of µ.Definition 2.3 (see [7]).Let f be a mapping from X into Y , and let µ ∈ L X and ν ∈ L Y .The L-subsets f (µ) ∈ L Y and f −1 (ν) ∈ L X , defined by, for all y ∈ Y , and for all x ∈ X, are called, respectively, the image of µ under f and the preimage of ν under f .Definition 2.4 (see [7]).Let µ ∈ L M .Then µ is said to be an Saying µ is a left L-module means that µ is an L-submodule of some left module M over a ring R. The set of all L-submodules of M is denoted by L(M).
Remark 2.5.We note from [7] that if µ,η ∈ L(M), then µ + η ∈ L(M).Also if µ i ∈ L(M), i ∈ I, then i∈I µ i ∈ L(M).From [6], we see that µ ∈ L(M) if and only if µ a is an R-module for all a ∈ L. Definition 2.6 (see [7]).Let M and N be R-modules and let µ ∈ L(M) and ν ∈ L(N).An isomorphism f of M onto N is called a weak isomorphism of µ into ν if f (µ) ⊆ ν.If f is a weak isomorphism of µ into ν, then say that µ is weakly isomorphic to ν and write µ ν.
Definition 2.7 (see [4]).Let A i , i ∈ Z, be R-modules and let µ i ∈ L(A i ).Suppose that Definition 2.9 (see [4]).Let be two isomorphic short exact sequences of R-modules with the given isomorphisms φ, ψ, and ξ.
Definition 2.11 (see [4]).Let A and B be two R-modules, µ ∈ L(A), η ∈ L(B).Consider the direct sum A ⊕ B. Extend the definition of µ and η to A ⊕ B to get µ and η in L(A ⊕ B) as follows: Therefore µ + η is in fact a direct sum and is denoted by µ ⊕ η.

Injective L-modules
The concept of free fuzzy modules was introduced by Muganda [8], which is later generalized to that of free L-modules (cf.[7]).Zahedi and Ameri [9] introduced the concepts of fuzzy projective and injective modules.In this section, we give an alternate definition for injective L-modules and prove that a direct sum of L-modules is injective if and only if each summand in the sum is injective.Also we prove that if µ ∈ L(J) is an injective Lmodule, and if 0 Definition 3.1.Let J be an injective R-module and let µ ∈ L(J).Then µ is said to be an injective L-module if for R-modules A, B, and η ∈ L(A), ν ∈ L(B), g any monomorphism from A to B such that g(η) = ν on g(A), and f : A → J any R-module homomorphism such that f (η) = µ on f (A), there exists an R-module homomorphism h : B → J such that hg = f and h(ν) ⊆ µ.
From the crisp module theory, it is known that an R-module J is injective if and only if every short exact sequence 0 An analogous result exists in the case of L-modules.
To prove this we need the following theorem.
Proof.Since J is injective, it is well known that any short exact sequence 0 is an injective L-module, and since f (µ) = ν on f (J), from the definition, we get h(ν) ⊆ µ.Thus there exists a homomorphism h : B → J such that h f = I J and h(ν) ⊆ µ.Then, by the above theorem, 0 In the crisp theory, we have the theorem that a direct sum of modules is injective if and only if each summand is injective.The same is true in the fuzzy case.

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.