Almost Injective Semimodules

In this work, injective semimodule has been generalized to almost -injective semimodule. The aim of this research is to study the basic properties of the concept almost-injective semimodules. The semimodule ℳ is called almost 𝒩 -injective semimodule if, for each subsemimodule A of 𝒩 and each homomorphism 𝜉 : A → ℳ , either there exists a homomorphism 𝜁 such that 𝜁𝑖 = 𝜉 . Or there exists a homomorphism 𝛾 : ℳ → Y such that 𝛾𝜉 = 𝜋 , where Y is nonzero direct summand of 𝒩 , and 𝜋 is the projection map. A semimodule ℳ is almost injective semimodule if it is almost injective relative to all semimodules. Every injective semimodule is almost injective semimodule, if ℳ is almost 𝒩 –injective semimodule and 𝒩 is simple, then ℳ is 𝒩 - injective. In addition, some related concepts it have been studied and investigated as well.


1.Introduction
In 1989 Baba introduced the concept "almost N-injective module" and he explained some properties of this concept, some related concepts were discussed in [1].
Lately, Singh 2016 , some conditions have been set under which U is almost V-injective module [2], which is generalization of the Baba's result.As regards semimodule, in 1998 Huda Althani gaves an equivalent definition of injective semimodules, which reduces to that in ISSN: 0067-2904 module theory.Also she studied some characterization of injective semimodules [3], later other authors discussed some generalizations of injective semimodules [4 ], [ 5] and [6].In this work, the concept of injective semimodule has been extended to generalization, almostinjective semimodule.Some characterizations of this notion and some concepts related to it will be discussed.Also, the conditions which want to get properties and attributes similar or related to the case in modules will be discussed.
By this paper, R will be denote a commutative semiring with identity 1≠0.ℳ will be a semimodule over R. Almost Ɲinjective semimodule was introduced and investigated.This paper has been organized as follows: Section 2,The main contributions have been introduced.In Section 3, The concluding remarks of this work are given.
A left R-semimodule is a commutative monoid (ℳ, +) with additive identity 0 ℳ and a function R× ℳ ⟶ ℳ denoted by (r, m) ↦ r m which is called scalar multiplication, such the following conditions hold, ∀ r, r ʹ, rʹʹ∈ R and m , m'∈ ℳ.(1) (r rʹ ) m = r (rʹm).(2) r (m+ mʹ) = r m+ r mʹ.(3) (r+ r' )m = r m+ rʹ m .(4)r 0 ℳ = 0 ℳ = 0Rm.The semimodule ℳ is called unitary if the condition 1m = m, for all m in ℳ, [7].A nonempty subset U of a left R-semimodule ℳ is called subsemimodule if U is closed under addition and scalar multiplication, denoted by U ≤ ℳ, [7].A subsemimodule U of ℳ is called subtractive subsemimodule if for each x, y ∈ ℳ, that x+ y , x ∈ U implies y ∈ U. A semimodule ℳ is called subtractive semimodule if it has only subtractive subsemimodules [7].A semimodule ℳ is said to be semisubtractive, if for any x, y ∈ ℳ there is z ∈ ℳ such that x+ z = y or some t ∈ ℳ such that y + t = x [4].An element m of left R-semimodule ℳ is called cancellable if m+ x= m +y implies that x=y.The R-semimodule ℳ is cancellative if and only if every element of ℳ is cancellable [9].A semimodule ℳ is said to be direct sum of subsemimodules K and L denoted by ℳ =K⨁L if each m ∈ ℳ uniquely written as m =k +l where x ∈ K and l ∈ L, then K and L are said to be direct summand of ℳ, denoted by K ≤ ⨁ ℳ [6].An R-semimodule  is called ℳ-injective ( is injective relative to ℳ ) if for every subsemimodule U of ℳ and any R-homomorphism from U to  can be extended to .The semimodule  is said to be injective if it is injective relative to every left R-semimodule [4].A nonzero R-semimodule ℳ is called simple if ℳ has no nonzero proper subsemimodule [10].A subsemimodule U of ℳ is called large (essential) if U∩ K ≠0 for every nonzero subsemimodule K of ℳ, denoted by U ≤  ℳ [11].A subsemimodule L of R-semimodule ℳ is called fully invariant if for each endomorphism f : ℳ → ℳ, then f(L)⊆  [10].An R-semimodule ℳ is called uniform if any subsemimodule L of ℳ is essential in ℳ [6].A semimodule ℳ is said to be indecomposable if it is nonzero and the direct summands of it are only {0} and it self, [6].A subsemimodule U of ℳ is called closed if it has no proper essential extension in ℳ, [6].Let ℳ be an Rsemimodule, U and V are subsemimodules of ℳ, U is called intersection complement (shortly, complement) of V if U∩V=0 and U is maximal with respect to this property.U and V are said to be mutually complement if they are complement of each other [6].It is clear that K is closed subsemimodule if and only if K is a complement in ℳ [6].An R-semimodule ℳ is called CS-semimodule if every subsemimodule of ℳ is large in direct summand of ℳ, equivalently, every closed subsemimodule of ℳ is direct summand of it [12].Let ℳ be an R-semimodule and L be a subsemimodule of ℳ, then ℳ is said to be maximal essential extension of L if  is proper extension of ℳ, then  is not essential extension of L [6].An Rsemimodule  is said to be injective hull of semimodule ℳ, if  is injective and it is essential extension of ℳ [6].

Almost Injective Semimodules
In this section, the concept ℳ is almost -injective semimodule will be presented as generalization of injective semimodule as well as investigating some properties of this notion.The following proposition is a characterization of almost -injective semimodule.It is well-known, every module over a ring has an injective hull, but this is not hold in general, for semimodules over a semiring, [6].(2) From ( 1) A semimodule ℳ is said to satisfy C3-condition, if for any subsemimodules U,Ѵ which are direct summand of ℳ such that U∩Ѵ=0, then U⨁Ѵ is also a direct summand of ℳ [6].Proposition 2.17: Let ℳ be semisubtractive, cancellative, quasi-continuous semimodule and  be any semimodule.Then ℳ is almost  -injective semimodule if and only if for any Rhomomorphism : U→ ℳ has no extension from  to ℳ, where U is subsemimodule of , then: (1) There exist decompositions Proof: Assume that ℳ is almost  -injective semimodule.(1) By Proposition 2.16, there is decomposition  =Y ⨁Z with Y ≠ 0 and R-homomorphism : ℳ → Y such that  is monomorphism on V ∩ Y, W = (V ∩ Y) and U = ker() are summands of ℳ, and  (v) = (v) for any v in V.As W and U are complements of each other and ℳ satisfies C3-condition, then ℳ = W ⨁ U, and (ℳ) = (W).
In [8] the concept total quotient semiring which is R-semimodule( quotient field) is studied and discussed.

Conclusion
Semirings are moved from rings however at the same time there are important difference between them.A semimodule Ӎ over semiring R is defined similarly in module over ring.Every module over ring is semimodule over semiring but the converse not true.In this work, some remarks and lemmas that help us to avoid some problems which are encountered were developed and discussed by using some properties of semimodule.

Corollary 2 .
19: Let D be a commutative semidomain and Q be quotient field, then D is almost QD -injective semimodule.Proof: Let : V→ D has no extension from Q to D, where V is maximal subsemimodule of QD, then Q ≠ D, since QD is injective, there exists : QD → Q D extension of .Let Y=  −1 (D), then Y = qD for some q ∈ Q such that () = 1.It is clear that V ⊆ Y.  (Y) = D, By maximality V = Y and from Corollary 2.18 (1) we have D is almost Q D -injective semimodule.