Almost Projective Semimodules

: The basis of this paper is to study the concept of almost projective semimodules as a generalization of projective semimodules. Some of its characteristics have been discussed, as well as some results have been generalized from projective semimodules.


Introduction:
Module theory is one of the important branches of algebra.In recent years, interest in semimodules has appeared as a generalization and extension of the module.The Projective module plays a useful role in the study of the category of modules.Some authors generalize projective modules for semimodule.In this paper projective semimodule has been expanded to almost projective semimodule.Every module over the ring is semimodule but the converse is not true.The major objective of this work is to represent and investigate the dual notion of almost injective semimodules.In 1 , the authors study the concept of an almost projective module and they discussed the concept of almost injective modules as a dual of almost projective modules.In this paper, almost projective modules have been expanded for semimodules taking into account the differences between modules and semimodules, which are mainly derived from their definitions.Almost projective semimodule which is a dual of almost Injective semimodule studied by K. Aljebory and A. Alhossaini 2 .If Ӎ and Ɲ are two semimodule, a semimodule Ӎ is said to be almost Ɲ-projective, if for each epimorphism ϵ: Ɲ ⟶ X where X is any semimodule and every homomorphism : Ӎ ⟶ X, either there is a homomorphism : Ӎ ⟶ Ɲ such that ϵ  = , or, there is a homomorphism γ: Y ⟶ Ӎ where Y is a nonzero direct summand of Ɲ such that  = ϵ Y where Y is the injection map from Y into Ɲ.A semimodule Ӎ is called almost-projective if it is almost Ɲ-projective for every finitely generated Ȑ-semimodule Ɲ.Some characterizations of this notion have been discussed.Throughout this work, Ȑ will be semiring with identity and Ӎ be a unitary Ȑ-semimodule.A semiring Ȑ is a nonempty set with both binary operations addition and multiplication such that; (Ȑ, +) is an abelian monoid with identity element 0; (Ȑ,•) is a monoid with identity 1; the multiplication distributives over the addition and 0 ŗ = ŗ 0, ∀ŗ ∈ Ȑ 3 .Clearly, every ring is a semiring but the converse is not true, a trivial example of a semiring that is not a ring is (ℕ, +, •).A left Ȑ-semimodule Ӎ is a commutative monoid (Ӎ,+) together with operation Ȑ × Ӎ → Ӎ; defined by (ȿɱ)⟼ȿɱ such that ∀ȿ, ᶉ ∈ Ȑ and ɱ, ɳ∈Ӎ, ȿ(ɱ + ɳ) = ȿɱ + ȿɳ; (ȿ + ᶉ)ɱ = ȿɱ + ᶉɱ; (ȿᶉ)ɱ = ȿ(ᶉɱ); 0 Ȑ ɱ = 0 Ӎ = ȿ0 Ӎ and 1Ȑɱ =ɱ if there is 1 Ȑ , Ӎ is called unitary semimodule 3 (in the same way the right semimodule is defined).A nonempty subset Z of a semimodule Ӎ is called a sub semi-module if Z is closed under addition and scalar multiplication.An Ȑ-subsemimodule Ӿ of Ӎ is called subtractive if, ∀ȶ, ȿ ∈ Ӎ, ȶ + ȿ ∈ Ӿ, ȶ ∈ Ӿ implies ȿ ∈ Ӿ.An Ȑsemimodule Ӎ is said to be subtractive if it has only subtractive subsemimodules 3 .An Ȑ-semimodule Ӎ is called semisubtractive if for any ɱ, ɳ ∈ Ӎ , there is ȿ ∈ Ӎ such that ɱ + ȿ = ɳ, or some ȶ ∈ Ӎ that ɳ + ȶ = ɱ 4 .This research branches into two parts, in part 1, some definitions and remarks that have needed in the paper and found in some literature have been represented.Part 2, deals with this topic with some of its qualities and linked it to other concepts.Preliminaries: Concepts related to this work are defined in this section.Definition 1: 5 An element ɱ of semimodule Ӎ is said to be cancellable if ɱ + ɳ = ɱ + ȿ, then ɳ = ȿ.If each element in Ӎ is cancellable, then Ӎ is called a cancellative semimodule.Definition 2: 6 Let Ӿ and Ұ be two subsemimodules of Ȑ-semimodule Ӎ. Ӎ is called the direct sum of Ӿ and Ұ, if each ɱ ∈Ӎ has uniquely written as ɱ =  +  where  ∈ Ӿ,  ∈ Ұ, denoted by Ӎ = Ӿ⨁Ұ, each Ӿ and Ұ is said to be a direct summand of Ӎ. Remark 1: If Ӎ is a direct sum of sub semimodules Ӿ and Ұ, then Ӿ ∩ Ұ =0 and Ӎ = Ӿ + Ұ, but the converse is not true.For  4 ={0, 1, , }, (  4 ,+) is monoid with addition defined by; ǩ + ǩ = ǩ , ǩ + ɦ =1 and for all ǩ , ɦ ∈  4 ,  4 is semimodule where ={0,1}with 1+1=1.(,+,•) is a semiring and it is called Boolean semiring.Take a subsemimodule Ӿ = {0, 1, }, then Ӿ = {0, 1} + {0, } and {0,1} ∩ {0, } = {0} but 1 can be written by, 1 = 1 + 0 and 1 = 1 + a this means 1 has no uniquely representation.The converse is true if the semimodule Ӎ has the conditions; semi-sub tractive and cancellative 6 .Knowing that it is always satisfied in the module.Remark 2: 7 An Ȑ-semimodule Τ is called free if and only if Τ is isomorphic to a direct sum of copies of a semiring Ȑ. Remark 3: The Ȑ-semimodule Ȑ is free (by Remark 2).Definition 3: 6 An Ȑ-semimodule Ӎ is said to be an Ɲ-projective semimodule, if for each Ȑepimorphism ∅: Ɲ → Ӿ and each Ȑ-homomorphism : Ӎ → Ӿ, there is a homomorphism : Ӎ → Ɲ such that ∅= .A semimodule Ӎ is called projective if it is projective relative to every Ȑ-semimodule.Proposition 1: 7 Every free semimodule is projective.In particular, every semiring with identity is projective over itself.Definition 4: 8 A subsemimodule Ӿ of Ӎ is said to be fully invariant if for each endomorphism ∅ of Ӎ, then ∅(Ӿ) ⊆ Ӿ. Remark 4: 6 The subtractive subsemimodules of ℕ over itself are of the form  ℕ,  ∈ ℕ.Where ℕ is the set of natural numbers.Example 1: Every subtractive subsemimodule of ℕ as ℕ-semimodule is fully invariant.Since the subtractive subsemimodules of ℕ are of the form k ℕ, k∈ ℕ let ∅: ℕ → ℕ, then ∅(ℕ) = ∅(ℕ) ⊆ ℕ.Definition 5: 9 A semimodule Ӎ is called to be indecomposable if the direct summands of it are only {0} and Ӎ.

Almost projective semimodules
In this section, a new generalization of projective semimodules is introduced and some characterizations of this notion are discussed.Definition 6: A semimodule Ӎ is said to be almost Ɲ-projective, if for each epimorphism α: Ɲ ⟶ Ӿ and every homomorphism : Ӎ ⟶ Ӿ, either there is : Ӎ ⟶ Ɲ such that α  = , or there is a homomorphism γ: Ұ ⟶ Ӎ where Ұ is a nonzero direct summand of Ɲ such that  = αҰ, where Ұ: Ұ ⟶ Ɲ is the injection map.This means either Fig. 1a or Fig. 1b commutes.(2) Every projective semimodule is almostprojective.Remark 6: It is well known that every projective module is a direct summand of a free module, this is not true in semimodule as Example (2.3) in 10 .

Open problem:
Under what conditions, ʽʽƝ-projectiveʼʼ in proposition 12, can be replaced by an almost Ɲ-projective semimodule, such that it is valid?

Conclusion:
In this work, a generalization of projective semimodule was presented.This concept is studied in the module differently from what was dealt with in this work.Some basic properties of this notion were discussed.Since every module is semimodule but the converse is not true, in some results, some conditions were added to achieve them.

Figure 2 .
Figure 2. Summand of Ӎ Where  Ұ and Ұ are the projection and injection maps respectively and Ӿ is any semimodule.Since Ӎ is almost Ɲprojective semimodule, then either there is :Ӎ ⟶ Ɲ such that   =   Ұ Where : Ɲ ⟶ Ӿ is an epimorphism, or there is : Ṵ ⟶ Ӎ such that   Ұ =  Ṵ where Ṵ is a nonzero direct summand of Ɲ, in the first diagram define  ′ =  Ұ , then  ′ =  Ұ =   Ұ  Ұ = .Now from the second