Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings

e LA-module is a nonassociative structure that extends modules over a nonassociative ring known as left almost rings (LArings). Because of peculiar characteristics of LA-ring and its inception into noncommutative and nonassociative theory, drew the attention of many researchers over the last decade. In this study, the ideas of projective and injective LA-modules, LA-vector space, as well as examples and ndings, are discussed. We construct a nontrivial example in which it is proved that if the LAmodule is not free, then it cannot be a projective LA-module. We also construct free LA-modules, create a split sequence in LAmodules, and show several outcomes that are connected to them. We have proved the projective basis theorem for LA-modules. Also, split sequences in projective and injective LA-modules are discussed with the help of various propositions and theorems.


Introduction
Kazim and Naseeruddin came up with the idea of left almost semigroups [1]. If a groupoid S meets (ab)c (cb)a for all a, b, c ∈ S, that is, left invertive law, then it is termed as LAsemigroup. Abel-Grassmann groupoid (abbreviated as AGgroupoid) is another name for this structure [2,3]. A structure that exists among a commutative semigroup and a groupoid is known as an AG-groupoid. LA-semigroup has been extended to left almost group (LA-group) by Kamran [4]. Groupoid G is referred to as a left almost group (LAgroup) if there will be a left identity e ∈ G (such that ea a for each a ∈ G), and for a ∈ G, there will be b in G implies that ba e. Also, left invertive law is true in G. e left almost group, despite having a nonassociative structure, bears an interesting resemblance to a commutative group.
Many researchers produced various valuable results for LA-semigroups and LA-groups due to nonsmooth structure development. With these ideas, the theory of the left almost ring was introduced [5]. e byproduct of LA-semigroup and LA-group is the left almost ring (LA-ring). Due to its unique properties, it has gradually developed as a useful nonassociative class with a decent contribution to nonassociative ring theory. A nonempty set R with at least two elements is a LA-ring if (R, +) is LA-group, (R, ·) is a LAsemigroup, and both left and right distributive laws are satis ed. For example, we can get LA-ring (R, ⊕, ·) from commutative ring (R, +, ·) by declaring a⊕b b − a for all a, b ∈ R and a · b is same as it was in the ring.
Shah and Rehman extended the study of LA-rings in [6]. Shah and Rehman [7] examined certain features of LA-rings using their ideals, and as a result, the ideal theory can be a good place to start looking into fuzzy sets and intuitionistic fuzzy sets. Mace4 has been used for certain computational tasks, and interesting and useful LA-ring properties have been examined [8]. In [6], the concept of a commutative semigroup ring is generalized using both the LA-semigroup and the LA-ring. Moreover, Shah and Rehman also develop the concept of a LA-module, which is a nonabelian nonassociative structure that is closer to an abelian group. As a result, studying this algebraic structure is quite similar to studying modules which are fundamentally abelian groups. Shah et al. [9] have done additional work in the subject of LA-modules, establishing various isomorphism theorems and direct sum of LA-module results. Alghamdi and Sahraoui [10] developed and built a tensor product of two LAmodules lately, extending simple conclusions from ordinary tensor to the new scenario. In [11], by defining exact sequences, Asima Razzaque et al. added to the study of LAmodules. Shah et al. [12] presented a complete survey and advances of the existing literature of nonassociative and noncommutative rings, as well as a list of some of their varied applications in diverse fields. Recently, Rehman et al. [13] introduced the concept of neutrosophic LA-rings. In 2020, Razzaque et al. [14], worked on soft LA-modules by defining projective soft LA-modules, free soft LA-modules, split sequence in soft LA-modules, and establish various results on projective and injective soft LA-modules. Abulebda in [15] discussed the uniformly primal submodule over noncommutative ring and generalized the prime avoidance theorem for modules over noncommutative rings to the uniformly primal avoidance theorem for modules. In [16], Groenewald worked on weakly prime and weakly 2-absorbing modules over noncommutative rings. He introduced a weakly m-system and characterized the weakly prime radical in terms of weakly m-system. Putman and Sam in [17] introduced VIC-modules over noncommutative rings. ey proved a twisted homology stability for GLn(R) with R a finite noncommutative ring. Nonassociative ring structure was enriched by introducing the hyperstructures. Rehman et al. in 2017 [18] have given the concept of LAhyperrings.
rough their hyperideals and hypersystems, they investigate various important characterizations of LAhyperrings. Massouros and Yaqoob [19] presented the study of algebraic structures, left/right almost groups, and hypergroups equipped with the inverted associativity axiom, and they analyzed the algebraic properties of these special nonassociative hyperstructures. We refer readers to see if they want to learn more about LA-rings [9,[20][21][22][23].
Some further developments in the field of modules were done by Ansari and Habib in [24] by defining their graphs over rings. ey investigate the relationship between the graph-theoretic properties and algebraic properties of modules. Moreover, Madhvi and Talebi defined the small intersection graph of submodules of modules [25]. In addition, Abbasi et al. presented a new graph connected with modules over commutative rings in [26]. ey look at the connection between some algebraic features of modules and the graphs that go with them. For the completeness of the special subgraphs, they gave a topological characterization. Furthermore, in 2017, Rajkhowa and Saikia worked on the graphs of noncommutative rings by defining the total directed graphs of noncommutative rings [27]. For more study of graphs of rings and graphs of modules over rings, we advised the readers to study [28][29][30][31][32][33][34].
In this work, we introduce the concepts of projective and injective LA-modules, LA-vector space, as well as examples and findings, over nonassociative and noncommutative rings. We construct a nontrivial example in which it is proved that if the LA-module is not free, then it cannot be a projective LA-module. We also construct free LA-modules, create a split sequence in LA-modules, and show several outcomes that are connected to them. We have proved the projective basis theorem for LA-modules. Also, split sequences in projective and injective LA-modules are discussed with the help of various propositions and theorems.

Background
In 2011, Shah et al., [9] promoted the notion of LA-module over an LA-ring defined in [6] and further established the substructures, operations on substructures, and quotient of an LA-module by its LA-submodule. ey also indicated the nonsimilarity of an LA-module to the usual notion of a module over a commutative ring. Shah et al. [9] have done more work on LA-modules, proving numerous isomorphism theorems and establishing a direct sum of LA-module findings. Alghamdi and Sahraoui [10] recently developed and constructed a tensor product of two LA-modules, extending simple conclusions from ordinary tensor to the new scenario. In [11], Asima Razzaque et al. contributed to the study of LA-modules by defining exact sequences and split sequences.
In the following, we will go over some basic definitions and findings related to the LA-modules.
Definition 1 (see [6]). Let (R, +, .) be LA-ring having left identity e. (M, +) an LA-group is called LA-module over R, For every a, b ∈ R, m, n ∈ M. R M or simply M is the abbreviation for the left R LAmodule. M R denotes the right R LA-module, which can be defined similarly.
Shah and Rehman [6] developed a nontrivial example of LA-module in the following example. e following example shows that every LA-module is not a module.
Definition 2 (see [9]). Consider the left R LA-module M. en, an abelian LA-subgroup N over LA-ring R is left R LA-submodule, if the condition RN⊆N holds, which means rn ∈ N for each r ∈ R, n ∈ N.
Theorem 1 (see [19]). A ∩ B is LA-submodule of M, where A and B are the LA-submodules of an LA-module M.
Definition 3 (see [9]). φ: M ⟶ N is called LA-module homomorphism if for all r∈R and m, n∈M, where M and N are LA-modules over LA-ring R.
Theorem 2 (see [9]). e following statements hold if φ: M ⟶ N is LA-module homomorphism: [35]). ∩ i∈I M i and i∈I M i are submodules of M, where M i |i ∈ I is a nonempty family of submodules.
Definition 4. In [35], readers can see the definition of a short exact sequence.

Proposition 2.
In [36], readers are referred to a proposition in which the relationship between exact sequence, monomorphism, epimorphism, and isomorphism of the modules is developed.

Main Results
We divide our work into two sections and look into a number of significant findings that are backed up with examples. roughout the paper, R denotes an LA-ring.

Projective LA-Module.
is section begins with a definition of the projective LA-module as well as an example.
en, M is projective LA-module. In Figure 1, if R LAmodules and LA-homomorphisms have an exact row, then there is an R LA-homomorphism g: M ⟶ A which results in the completed diagram commutative which means αg � f.
To construct the example of a projective LA-module, first, we need to define LA-vector space.
where f ∈ F and v ∈ V satisfy the following conditions: It is easy to verify the (ii), (iii), and (v) property. Here, we only prove the (iv) property of LA-vector space.
is implies s 1 , s 2 , . . . , s n are linearly independent. Now let f 1 , f 2 , . . . , f n ∈ F and s 1 , s 2 , . . . , s n ∈ S. en, f 1 s 1 + f 2 s 2 + · · · + f n s n is linear combination of elements of S whose coefficients are from LA-field F. erefore, S is free basis for F[S] as an LA-vector space over LA-field F. Example 3. An LA-vector space over an LA-field F is free F LA-module, so is a projective LA-module.

Definition 7.
A left R LA-module F is called free left R LAmodule on a basis X ≠ ϕ, if there will be a map α: X ⟶ F such that the given map f: Unique LA-homomorphism g: F ⟶ A is said to extend the map f: X ⟶ A.
R-homomorphism is referred to as R LA-homomorphism in this study. Proof. Suppose F is a free LA-module having basis X.
In Figure 2, R LA-modules and R-homomorphism have the row exact. Let x ∈ X. en, f(x) ∈ B and as α is onto, so there exists a ∈ A then α(a) � f(x). Define g: X ⟶ A as g(x) � a and extend the function g: F ⟶ A, and here, it is clear On the other hand, if LA-module is not free, then it will not be projective LA-module. is remark is justified in the next example.  Figure 4 has an exact row. M is as projective LA-module, so there will be homomorphism g: Proof. Assume for each j ∈ J, M j is a projective LA-module. Figure 5 has an exact row. e homomorphism fi j : M j ⟶ B for every j ∈ J and M j is the projective LAmodule. Hence, there is a homomorphism g j : M j ⟶ A such that αg j � fi j . Now, define g: M ⟶ A by g(x) � j g j π j (x), for x ∈ M (see Figure 6).
It is obvious the right side sum is finite. erefore, g is the homomorphism. Let x ∈ M, αg(x) � α( j g j π j (x)) � j αg j π j (x) � j fi j π j (x) � f( j i j π j (x)) � f(( j i j π j ) (x)) � f(x). It shows αg � f. It is clear M is projective LAmodule. Conversely, let M a projective LA-module. We have Figure 7 having an exact row.
For any j ∈ J, a homomorphism fπ j : M ⟶ B where M is a projective LA-module. ere will be a homomorphism g: M ⟶ A so that αg � fπ j . Now, let take g j � gi j which is a homomorphism from M j ⟶ A, then LA-modules and homomorphisms is called splits or split sequence of LA-modules. If any of the statement is true,  Proof. First, M is projective LA-module, F is free R LA-module having the basis x i : i ∈ I , and ϕ: F ⟶ M is epimorphism. As M a projective LA-module over R, there will be homomorphism c: M ⟶ F so that ϕc � ⊥ M . Now, for any i ∈ I, define Figure 9). It is clear that ϕ i are well-defined. As F is free LAmodule on x i : i ∈ I , ϕ i are clearly LA-homomorphisms. Since c(m) is finite sum i∈I r i x i , ϕ i (m) � 0 for almost every i.
Hence, (i) and (ii) are proved. Conversely, consider a subset x i : i ∈ I of M and a set ϕ i : M ⟶ R, i ∈ I of R LA-homomorphisms such that the conditions (i) and (ii) holds. Now, a set X � x i : i ∈ I of symbols that are indexed by the same set I and let F be a free LA-module having the basis X. ϕ: X ⟶ M defined by ϕ(x i ) � m i where i ∈ I spreads to a homomorphism ϕ: . By condition (i) satisfied by ϕ i , the right side of ( * ) is a finite sum. is implies c is LAhomomorphism. For m ∈ M, ϕc(m) � ϕ( i∈I ϕ i (m) Hence, ϕc � ⊥ M . erefore, ϕ an LA-epimorphism splits which follows that M becomes direct summand of the free LA-module F. erefore, M is projective LA-module. ere will be R LA-homomorphism g: B ⟶ I which results the completed diagram commutative which means gα � f (see Figure 10).
Alternatively, we can define injective LA-module as follows. (1) 0 { } LA-module is trivially an injective LA-module (2) Let K be an LA-field, then every K LA-vector space is an injective K LA-module Proposition 6. Let an injective R LA-module I. In the following figure having an exact row and fα � 0, there will be a homomorphism g: C ⟶ I which results the completed diagram commutative, that is, gβ � f (see Figure 11).
therefore, β − π � β. As I is an injective LA-module, there will be a homomorphism g: Proof. Since I � j∈J I j , there will be a homomorphism i j : I j ⟶ I and p j : I ⟶ I j which implies p j i j � ⊥ I j and p k i j � 0, the zero map if j ≠ k. Consider a monomorphism of R LA-modules α: A ⟶ B and assume that each I j is injective LA-module, considering diagram (see Figure 12) Assume f: A ⟶ I is homomorphism. en, p j f: A ⟶ I j is a homomorphism. As I j is injective, there will be a homomorphism g j : B ⟶ I j which implies g j α � p j f. Now, define g: B ⟶ I by g(b) � (g j (b)), b ∈ B. It follows g is a homomorphism, and for a ∈ A, g(α(a)) � (g j α(a)) � p j f(a) � f(a), which gives gα � f. is results I is an injective LA-module. Now, conversely, let I be injective LA-module. Let f j : A ⟶ I j be a homomorphism for any j ∈ J.
As I is injective LA-module, there will be a homomorphism g: B ⟶ I which implies gα � i j f j (see  Figure 13). It follows p j g: B ⟶ I j will be a homomorphism which means p j gα � p j i j f j � f j . Hence, I j is injective LA-module. Proof. In Figure 14, as I is injective LA-module, therefore, there will be a homomorphism g: A ⟶ I which implies gα � ⊥ I . It follows that exact sequence O ⟶ I ⟶ α A ⟶ β B ⟶ O splits.

Conclusion
Mathematics is becoming increasingly nonassociative and noncommutative. It is widely predicted that nonassociativity and noncommutativity will dominate mathematics and applied sciences in the coming years. In this paper, the study of LA-modules can be classified as a theoretical study in the development of nonassociative and noncommutative algebraic theory. e notions of split sequence, free LA-module, projective LA-modules, injective LA-modules, and their related features were discussed in relation to LA-modules. Further advancements in the study of LA-modules can be made by defining functors, pull back and push outs, and so on. In addition, LA-modules and its substructures can be defined in the study of neutrosophic sets and hyper structures. Also, neutrosophic graphs of these algebraic structures can be constructed. Moreover, graphs of LA-modules over nonassociative rings and nonassociative hyper structures can be defined. Furthermore, nonassociative rings and nonassociative hyper structures can be used in various decision-making procedures, and fuzzy theory and its applications can be extended to the medical sciences.

Data Availability
All data are available in the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Journal of Mathematics 7