Nearly Injective Semimodules

An injectivity in the category of semimodules over semiring was studied by many authors recently. On the other hand, the concept of injectivity, in the category of modules over ring, was generalized in many different directions. In particular, injective modules relative to preradical were some of those generalizations. As an analogue to module theory, in this paper, we introduce and investigate the notion of "injective semimodule relative to Jacobson radical (namely nearly injective semimodule)".


Introduction
Throughout this work, Ş stands for a commutative semiring with identity and a semimodule means a unitary left Ş-semimodule.An Ş-subsemimodule  of an Şsemimodule  is called subtractive if for all ,  ′ ∈ ,  + ′,  ∈  implies ′ ∈ , it is clear that 0 and  are subtractive Ş-subsemimodules of .An Ş-semimodule  is a subtractive Ş-semimodule if it has only subtractive subsemimodules [1].For convince all Ş-semimodules assumed in this work will be considered subtractive.
In 2000, nearly injective modules were discussed in [2] as a generalization of injective modules.A module  is called nearly injective, if for every monomorphism :  ⟶  (where  and  are two modules) and for each homomorphism :  ⟶ , there is a homomorphism ℎ:  ⟶  such that (ℎ ∘ )() − () ∈ (), ∀ ∈  where () is the Jacobson radical of the module , which is defined to be the intersection of all maximal submodules of .
As an analogue to the case in modules, nearly injective semimodule, is introduced in this work.An Ş-semimodule  is called nearly injective, if for every Şmonomorphism :  ⟶  (where  and  are two Ş-semimodules), each Ş-homomorphism :  ⟶ , there is an Ş-homomorphism ℎ:  ⟶  such that   ℎ =   , where   :  ⟶ /() is the natural epimorphism, and () is the Jacobson radical of .As in modules () is the intersection of all maximal Şsubsemimodules of .
Many properties and characterizations of nearly injectivity were investigated.The main results of this work are: It is shown that these semimodules are closed under arbitrary direct product, finite direct sum and direct summand.Nearly direct summand with proof that an Ş-semimodule  is nearly injective if and only if, it is a nearly direct summand of every extension of itself.Therefore the nearly split homomorphism with proof that an Ş-semimodule  is nearly injective if and only if, for each Şsemimodule , every Ş-monomorphism :  ⟶  is nearly split.
In addition to section 1, there are two sections.Section 2 consists the preliminaries that are needed in the investigations.Some of these were found in the literatures.In section 3, injective Ş-semimodule and their properties and characterizations were given.

2-Preliminaries
In this section same definitions, their properties and characterizations of these Şsemimodules needed in this work.

Definition 2.2 [3].
Let  be a subset of a left Ş-semimodule  then  is called subsemimodule of  if  is closed under addition and scalar multiplication.In this case it is denoted by  ↪ .
We denote to the Ş-semimodule that possess the three conditions, yoked, cancellative and subtractive by -semimodule.

Definition 2.6 [5]
An Ş-subsemimodule  of an Ş-semimodule  is called a direct summand of  if there exists Ş-subsemimodule  such that  = ⨁ and  is called a direct sum of  and .
For a homomorphism of Ş-semimodules  :  ⟶  we define: for some a, a'∈ }.

Definition 2.8 [3]
The (possibly infinite) sequence of Ş -semimodules → … is said to be: Proposition 2.9 [3] Let  and  be Ş-semimodules, then a homomorphism of Şsemimodules :  ⟶  is: 1. injective if and only if it is a monomorphism.2. surjective if and only if it is epimorphism and () ⊆  is subtractive.

𝜑(𝐴) is subtractive if and only if 𝜑(𝐴) = 𝐼𝑚(𝜑).
The following two lemmas and corollary had been proved for modules (see e.g.[6, pp.60-61]), but for semimodules they need extra conditions and then new converted proofs that were not found in the literatures.
Thus  ∈  +  () (in any case).The other direction is clear.Clearly if  is called ҟ-injective then  is injective.
(2) Clear.Since the product and coproduct coincide in the finite case, and by Proposition(2.16).Definition 2.18.[7].Let  be an Ş-semimodule.An Ş-subsemimodule  of  is called large (essential) Ş-subsemimodule of  if for every Ş-subsemimodule  of  , ⋂ = 0 implies  = 0, in this case we say that  is an essential extension of . is called maximal essential extension of  if whenever  is a proper extension of  then  is not an essential extension of .Note that we shall denote the statement ′′ is a large subsemimodule  of the Ş-semimodule  ′′ by  ≼ ℯ .Definition 2.19.An injective Ş-semimodule  is called minimal injective extension of an Ş-subsemimodule  if  is an extension of  and whenever  is a proper subsemimodule of  which contains  then  is not injective.
The following statement is true for any module, see for example [6, pp. 114], but for semimodules the subtractive condition is needed and we have to give a corresponding proof.Lemma 2.20.Let  be a subtractive Ş-semimodule.If  is a subsemimodule of  and  is a subsemimodule of  maximal with the property  ∩  = 0, then ⨁ ≼ ℯ .
Therefore ⨁ ≼ ℯ .□ Note: Analogue to the case in modules, such subsemimodule  (with the property given in the previous lemma will be called an intersection complement of  in  (shortly inco of  in ), see [6, D. 5.2.1].As in the proof above an inco of a given subsemimodule, always exists.Proposition 2.21.Let  be a -semimodule and contained in an injective Şsemimodule then  is injective if and only if it is a direct summand of every extension of itself.
Proof: Suppose that  is injective and  is a proper extension of , consider the following diagram.Where  is a monomorphism Such that ℎ =   ⟹ () is direct summand in  (Corollary (2.13)).
Since  is a monomorphism,  ≅ (), hence  is direct summand in .
Conversely, suppose that  is direct summand of every extension of it.Now, since  is contained in an injective Ş-semimodule then  has an injective extension Ş-semimodule.say .Thus  will be a direct summand of  and so will be injective by proposition (2.16).□ Proposition 2.22.Let  be an essential extension of  and let  an injective extension of  then the inclusion mapping of  into  can be extended to an embedding of  in .

Proof:
The same proof as in the case of modules (see [8, pp.41]).□ Proposition 2.23.Let  be an Ş-semimodule and  an injective extension of  then  has subsemimodule  which is a maximal essential extension of .

Proof:
The same proof as in the case of modules (see [8, pp.42]).□ The proof of the following Theorem (2.24) and Proposition (2.25) are similar to the case of modules, by considering extra condition that it is needed for semimodules (see [8, pp.43]).For completeness we give a full proof.Theorem 2.24.Let  be a -semimodule and contained in an injective Şsemimodule, then  is injective if and only if it has no proper essential extension.
Proof: Suppose that  is injective and let  be a proper extension of .Now, by Proposition (2.21)  is direct summand of , so it cannot be essential in .
Conversely, suppose that  has no proper essential extension.Now, let  be any extension of , and Let  be an inco of  in .(we assume that  is a proper extension of ).Then / ⊇ ( ⊕ )/ ≅  (by the isomorphism theorem see [6]).That is / is an extension of , and so by assumption  is not essential in / (( ⊕ )/ is not essential in / ).Then there exists  ⊆  such that / ∩ ( ⊕ )/ = 0 which implies  ∩ ( ⊕ ) = .
Since L is an arbitrary extension of W, thus W is a direct summand of any extension of it.
Proposition 2.25.Let  be a -semimodule which is contained in an injective Şsemimodule.If  is an Ş-semimodule contained in , then the following statements are equivalent: 1.  is an essential injective extension of . 2.  is a maximal essential extension of .
Let  be an injective extension of  contained in .Then  is an injective extension of , so  =  by Theorem (2.24) applied to .
Hence  is a minimal injective extension of .
(3) ⟹ (1) Assume,  is a minimal injective extension of .Now, by Proposition (2.23)  has a subsemimodule  which is a maximal essential extension of  and so injective it follows that  =  and (1) is established.□ An Ş-semimodule  satisfying the conditions of Proposition (2.25) is called an injective envelope (or injective hull) of  (if it exists), we use the notation () to stand for an injective envelope of  [9].

For every diagram,
Where i is the inclusion map and  is any homomorphism, there exists an Şhomomorphism ℎ:  ⟶  such that   ℎ =   , where   :  ⟶ /() is the natural epimorphism.

For every diagram with 𝐴 ≼ ℯ 𝐵,
There exists an Ş-homomorphism ℎ:  ⟶  such that   ℎ =   , where   :  ⟶ /() is the natural epimorphism.Then F is a free Ş-semimodule, and  can be considered as a subsemimodule of F.
Hence  is nearly injective Ş-semimodule.□ Now, we will study the direct product and the direct sum of nearly injective semimodules.The following propositions shows that this result is true in of nearly injective semimodules.(where  and  be two Ş-semimodules, and  be an Ş-homomorphism).
So, since each   is nearly injective Ş-semimodule, for any  ∈∧, there is an Şhomomorphism ℎ  :  ⟶   such that   ℎ   =      .1.If  is nearly injective then each   is nearly injective.

2.
If each   is nearly then  =⊕ ∈∧   is nearly injective where ∧ is a finite set.
Proof: (1) The proof is similar to the proof of necessity part of Proposition (3.12).
Hence by (ii)   ℎ 1  =   .Therefore  is a nearly direct summand of .□ The following theorem gives another characterization of nearly injective semimodules.

3 .
is a nearly direct summand of every injective extension of itself.4.  is a nearly direct summand of at least one injective extension of itself.Proof: (1) ⟹ (2) Assume that  is a nearly injective Ş-semimodule .Let  1 be any extension of , consider the following diagram with exact row, Since  is a nearly injective Ş-homomorphism, there exists an Ş-homomorphism ℎ:  1 ⟶  such that   ℎ =     ⟹   ℎ =   .That is  is nearly direct Summand of  1 by Proposition (3.16).