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

Razzaque, Asima; Rehman, Inayatur

doi:https://doi.org/10.1155/2022/2792450

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Volume 2022 | Article ID 2792450 | https://doi.org/10.1155/2022/2792450

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

Asima Razzaque¹and Inayatur Rehman²

Academic Editor: Gohar Ali

Received23 Apr 2022

Accepted21 Jun 2022

Published31 Aug 2022

Abstract

The LA-module is a nonassociative structure that extends modules over a nonassociative ring known as left almost rings (LA-rings). 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 findings, are discussed. 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.

1. Introduction

Kazim and Naseeruddin came up with the idea of left almost semigroups [1]. If a groupoid meets for all , that is, left invertive law, then it is termed as LA-semigroup. Abel-Grassmann groupoid (abbreviated as AG-groupoid) 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 is referred to as a left almost group (LA-group) if there will be a left identity (such that for each ), and for , there will be in implies that . Also, left invertive law is true in . The 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]. The 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 with at least two elements is a LA-ring if is LA-group, is a LA-semigroup, and both left and right distributive laws are satisfied. For example, we can get LA-ring from commutative ring by declaring for all and 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 LA-modules 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 LA-modules. 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. They proved a twisted homology stability for with a finite noncommutative ring. Nonassociative ring structure was enriched by introducing the hyperstructures. Rehman et al. in 2017 [18] have given the concept of LA-hyperrings. Through their hyperideals and hypersystems, they investigate various important characterizations of LA-hyperrings. 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, 20–23].

Some further developments in the field of modules were done by Ansari and Habib in [24] by defining their graphs over rings. They 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]. They 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 [28–34].

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.

2. 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. They 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 be LA-ring having left identity . an LA-group is called LA-module over , and a map is defined , where , satisfies(i)(ii)(iii)(iv)For every , .

or simply is the abbreviation for the left LA-module. denotes the right LA-module, which can be defined similarly.

Shah and Rehman [6] developed a nontrivial example of LA-module in the following example. The following example shows that every LA-module is not a module.

Example 1 (see [6]). Let commutative semigroup and be LA-ring with left identity. Then, , as defined by becomes an LA-module.

Definition 2 (see [9]). Consider the left LA-module . Then, an abelian LA-subgroup over LA-ring is left LA-submodule, if the condition holds, which means for each , .

Theorem 1 (see [19]). is LA-submodule of , where and are the LA-submodules of an LA-module .

Definition 3 (see [9]). is called LA-module homomorphism if for all and , where and are LA-modules over LA-ring .(i)(ii)

Theorem 2 (see [9]). The following statements hold if is LA-module homomorphism:(1) an LA-submodule of , where is LA-submodule of (2) an LA-submodule of , where is LA-submodule of

Proposition 1 (see [35]). and are submodules of , where 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.

Theorem 3. [37] Every free left module has a homomorphic image in every left module.

3. Main Results

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

3.1. Projective LA-Module

This section begins with a definition of the projective LA-module as well as an example.

Definition 5. Let denotes the left LA-module over . Then, is projective LA-module. In Figure 1, if LA-modules and LA-homomorphisms have an exact row, then there is an LA-homomorphism which results in the completed diagram commutative which means .
To construct the example of a projective LA-module, first, we need to define LA-vector space.

Definition 6. The triplet is an LA-vector space over an LA-field if defined as , where and satisfy the following conditions:(i) is an LA-group(ii)If and (iii)If and (iv)If and (v)For ,

Example 2. Let a commutative semigroup and is LA-field with left identity. Consider , which is obviously is an additive LA-group. Define the map by which is well-defined. It is easy to verify the (ii), (iii), and (v) property. Here, we only prove the (iv) property of LA-vector space. Consider . Since is an LA-field, then by ([38], Lemma 4), is true for all . Hence, . Thus, . Hence, it is proved.

Remark 1. Let be a commutative semigroup. It is easy to observe that becomes free basis for as LA-vector space over LA-field . Indeed, consider . Since and for each , so for each . This implies are linearly independent. Now let and . Then, is linear combination of elements of whose coefficients are from LA-field . Therefore, is free basis for as an LA-vector space over LA-field .

Example 3. An LA-vector space over an LA-field is free LA-module, so is a projective LA-module.

Definition 7. A left LA-module is called free left LA-module on a basis , if there will be a map such that the given map , where is any left LA-module, there exists a unique LA-homomorphism such that .
Unique LA-homomorphism is said to extend the map .

-homomorphism is referred to as LA-homomorphism in this study.

Theorem 4. Free LA-module implies projective LA-module.

Proof. Suppose is a free LA-module having basis .
In Figure 2, LA-modules and -homomorphism have the row exact. Let . Then, and as is onto, so there exists then . Define as and extend the function , and here, it is clear . This implies .

Remark 2. On the other hand, if LA-module is not free, then it will not be projective LA-module. This remark is justified in the next example.

Example 4. Following is the Cayley tables of LA-module over
From tables, it is can be seen that additive LA-group becomes LA-module. If is the order of the finite LA-group , then , and also, we can observe from the table that 2 is the zero element of . Choose an element we see that , also , and likewise, it can be proved that for all , . Where can never be a part of basis. Thus, is not a free LA-module. So is not a projective LA-module.

Proposition 3. Figure 3 LA-modules and LA-homomorphisms have exact row, . This gives the homomorphism so that .

Proof. If and be the induced homomorphism by .From , this follows that . So a homomorphism is induced by , if becomes inclusion map, so and also . Figure 4 has an exact row. is as projective LA-module, so there will be homomorphism so . On the other hand, . This implies .

Proposition 4. is projective LA-module iff is projective LA-module for each .

Proof. Assume for each , is a projective LA-module. Figure 5 has an exact row. The homomorphism for every and is the projective LA-module. Hence, there is a homomorphism such that . Now, define by , for (see Figure 6).
It is obvious the right side sum is finite. Therefore, is the homomorphism. Let , . It shows . It is clear is projective LA-module. Conversely, let a projective LA-module. We have Figure 7 having an exact row.
For any , a homomorphism where is a projective LA-module. There will be a homomorphism so that . Now, let take which is a homomorphism from , then . Hence, is projective LA-module.

Definition 8. A short exact sequence of LA-modules and homomorphisms is called splits or split sequence of LA-modules. If any of the statement is true,(i)A homomorphism exists if (ii)A homomorphism exists if (iii) is a direct summand of

Lemma 1. Homomorphic image of projective LA-module is LA-module.

Proof. Straightforward by Theorems 3 and 4.

Theorem 5. is projective LA-module iff every exact sequence splits.

Proof. Consider an exact sequence (see Figure 8), by definition of projective LA-module, we have homomorphism implies which results that the split sequence. Conversely, let the sequence splits as every LA-module is a homomorphic image of free LA-module. is epimorphism where is free LA-module. If is the kernel , we get exact sequence which splits by the supposition. Hence, . becomes free projective LA-module, and this implies and are projective.

Proposition 5 (Projective basis theorem for LA-module). An LA-module is projective iff there will be a subset of and of LA-homomorphisms, then(i)For any , for almost all .(ii)For any , . Then, is generated by .

Proof. First, is projective LA-module, is free LA-module having the basis , and is epimorphism. As a projective LA-module over , there will be homomorphism so that . Now, for any , define by where , (see Figure 9). It is clear that are well-defined. As is free LA-module on , are clearly LA-homomorphisms. Since is finite sum , for almost every . For , define . If , then. Hence, and are proved. Conversely, consider a subset of and a set of LA-homomorphisms such that the conditions (i) and (ii) holds. Now, a set of symbols that are indexed by the same set and let be a free LA-module having the basis . defined by where spreads to a homomorphism . If , then which gives the result that is epimorphism. Define by . By condition (i) satisfied by , the right side of is a finite sum. This implies is LA-homomorphism. For , by condition (ii). Hence, . Therefore, an LA-epimorphism splits which follows that becomes direct summand of the free LA-module . Therefore, is projective LA-module.

3.2. Injective LA-Module

We define injective LA-modules and establish several essential conclusions in this part.

Definition 9. Let denotes the left LA-module over . Then, is injective LA-module, if in the figure LA-modules and LA-homomorphisms having exact row.
There will be LA-homomorphism which results the completed diagram commutative which means (see Figure 10).
Alternatively, we can define injective LA-module as follows.

Definition 10. A left LA-module over is injective, if is LA-submodule of some other left LA-module . Then, there will be another LA-submodule of so the internal direct sum of and is , that is, and .

Example 5. In view of above definition, we have the following examples of injective LA-module.(1) LA-module is trivially an injective LA-module(2)Let be an LA-field, then every LA-vector space is an injective LA-module

Proposition 6. Let an injective LA-module . In the following figure having an exact row and , there will be a homomorphism which results the completed diagram commutative, that is, (see Figure 11).

Proof. Suppose . Then, induces a monomorphism , given by , where . Let , it follows is monomorphism. Also, but ; therefore, a homomorphism induced by , by , where . Let represents the natural projection. for every ; hence, and for all ; therefore, . As is an injective LA-module, there will be a homomorphism such that ; then, is proved.

Proposition 7. If ( direct product of ) and is a family of LA-modules, then an injective LA-module iff is injective.

Proof. Since , there will be a homomorphism and which implies and , the zero map if . Consider a monomorphism of LA-modules and assume that each is injective LA-module, considering diagram (see Figure 12)
Assume is homomorphism. Then, is a homomorphism. As is injective, there will be a homomorphism which implies . Now, define by . It follows is a homomorphism, and for , , which gives . This results is an injective LA-module. Now, conversely, let be injective LA-module. Let be a homomorphism for any .
As is injective LA-module, there will be a homomorphism which implies (see Figure 13). It follows will be a homomorphism which means . Hence, is injective LA-module.

Proposition 8. Every exact sequence splits if is injective LA-module.

Proof. In Figure 14, as is injective LA-module, therefore, there will be a homomorphism which implies . It follows that exact sequence splits.

4. 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. The 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

The authors declare that they have no conflicts of interest.

Acknowledgments

“This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Project no. GRANT305)”

References

M. A. Kazim and M. Naseerudin, “On almost semigroups,” Alig. Bulletin Math., vol. 2, pp. 1–7, 1972.
View at: Google Scholar
P. V. Protic and N. Stevanovic, “AG-test and some general properties of Abel-Grassmann’s groupoids,” Pure Mathematics and Applications, vol. 6, pp. 371–383, 1995.
View at: Google Scholar
P. V. Protic and N. Stevanovic, “The structural theorem for -groupoids,” Series Mathematics Informatics, vol. 10, pp. 25–33, 1995.
View at: Google Scholar
M. S. Kamran, “Conditions for LA-semigroups to resemble associative structures,” Quaid-i-Azam University, Islamabad, 1993, Ph.D. Thesis.
View at: Google Scholar
S. M. Yusuf, “On left almost ring,” in Proceedings of the 7th International Pure Mathematical Conference, Pakistan 2006.
View at: Google Scholar
T. Shah and I. Rehman, “On LA-rings of finitely non-zero functions,” International Journal of Contemporary Mathematical Sciences, vol. 5, pp. 209–222, 2010.
View at: Google Scholar
T. Shah and I. Rehman, “On characterizations of LA-rings through some properties of their ideals,” Southeast Asian Bulletin of Mathematics, vol. 36, pp. 695–705, 2012.
View at: Google Scholar
A. Razzaque, “Investigation of a class of non-associative rings and its study using soft sets,” Quaid-i-Azam University, Islamabad, 2016, Ph.D. Thesis.
View at: Google Scholar
T. Shah, M. Raees, and G. Ali, “On LA-modules,” International Journal of Contemporary Mathematical Sciences, vol. 6, pp. 999–1006, 2011.
View at: Google Scholar
A. M. Alghamdi and F. Sahraoui, “Tensor product of LA-modules,” International Mathematical Forum, vol. 9, no. 27, pp. 1309–1319, 2014.
View at: Google Scholar
A. Razzaque, I. Rehman, and K. P. Shum, “On soft LA-modules and exact sequences,” Italian Journal of Pure and Applied Mathematics, vol. 38, pp. 797–814, 2017.
View at: Google Scholar
S. T. Shah, A. Razzaque, I. Rehman, M. A. Gondal, M. I. Faraz, and K. P. Shum, “Literature survey on non-associative rings and developments,” European Journal of Pure and applied Mathematics, vol. 12, no. 2, pp. 370–408, 2019.
View at: Publisher Site | Google Scholar
I. Rehman, A. Razzaque, and M. I. Faraz, “Neutrosophic set approach to study the characteristic behavior of left almost rings,” Neutrosophic sets and Systems, vol. 46, pp. 24–36, 2021.
View at: Google Scholar
A. Razzaque, I. Rehman, M. Faraz, and K. P. Shum, “Characterization of generalized projective and injective soft modules,” Italian Journal of Pure and Applied Mathematics, vol. 43, pp. 802–817, 2020.
View at: Google Scholar
L. J. M. Abulebda, “Uniformly primal submodule over noncommutative ring,” Journal of Mathematics, vol. 2020, pp. 1–4, 2020.
View at: Publisher Site | Google Scholar
I. Rehman, “On generalized commutative rings and related structures,” Quaid-i-Azam University, Islamabad, 2011, Ph.D. Thesis.
View at: Google Scholar
A. Putman and S. V. Sam, “VIC-modules over noncommutative rings,” 2020, https://arxiv.org/pdf/2005.07246.pdf.
View at: Google Scholar
I. Rehman, N. Yaqoob, and S. Nawaz, “Hyperideals and hypersystems in LA-hyperrings,” Songklanakarin Journal of Science and Technology, vol. 39, no. 5, pp. 651–657, 2017.
View at: Google Scholar
C. G. Massouros and N. Yaqoob, “On the theory of Left/Right almost groups and hypergroups with their relevant enumerations,” Mathematics, vol. 9, no. 15, 2021.
View at: Publisher Site | Google Scholar
I. Rehman, M. Shah, T. Shah, and A. Razzaque, “On existence of non-associative LA-rings,” Analele Stiintifice ale Universitatii ovidius Constanta, vol. 21, no. 3, pp. 223–228, 2013.
View at: Google Scholar
T. Shah, N. Kausar, and I. Rehman, “Intuitionistic Fuzzy Normal subrings over a non-associative ring,” “Analele Universitatii” “Ovidius” “Constanta-Seria Matematica”, vol. 20, no. 1, pp. 369–386, 2012.
View at: Publisher Site | Google Scholar
N. J. Groenewald, “On weakly prime and weakly 2-absorbing modules over noncommutative rings,” Kyungpook Mathematical Journal, vol. 61, pp. 33–48, 2021.
View at: Google Scholar
M. Shah and T. Shah, “Some basic properties of LA-rings,” International Mathematical Forum, vol. 6, pp. 2195–2199, 2011.
View at: Google Scholar
H. Ansari-Toroghy and S. Habibi, “Domination number in the annihilating-submodule graph of modules over commutative rings,” International Electronic Journal of Algebra, vol. 30, pp. 203–216, 2021.
View at: Publisher Site | Google Scholar
L. Mahdavi and Y. Talebi, “On the small intersection graph of submodules of a module,” Algebraic Structures and Theit Applications, vol. 8, no. 1, pp. 117–130, 2021.
View at: Google Scholar
A. Abbasi, H. Roshan, and D. H. Lelekaami, “Associated graphs of modules over commutative rings,” Iranian Journal of Mathematical Sciences and Informatics, vol. 10, no. 1, pp. 45–58, 2015.
View at: Google Scholar
K. K. Rajkhowa and H. K. Saikia, “On total directed graphs of non-commutative rings,” AKCE International Journal of Graphs and Combinatorics, vol. 14, no. 3, pp. 261–268, 2017.
View at: Publisher Site | Google Scholar
S. Akbari, R. Nikandish, and M. J. Nikmehr, “Some results on the intersection graphs of ideals of rings,” Journal of Algebra and Its Applications, vol. 12, no. 04, Article ID 1250200, 2013.
View at: Publisher Site | Google Scholar
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, NY, USA, 1992.
I. Chakrabarty, S. Ghosh, T. K. Mukherjee, and M. K. Sen, “Intersection graphs of ideals of rings,” Discrete Mathematics, vol. 309, no. 17, pp. 5381–5392, 2009.
View at: Publisher Site | Google Scholar
D. F. Anderson and A. Badawi, “The total graph of a commutative ring,” Journal of Algebra, vol. 320, no. 7, pp. 2706–2719, 2008.
View at: Publisher Site | Google Scholar
A. Abbasi and S. Habibi, “The total graph a module over a commutative ring with respect to proper submodules,” Journal of Algebra and Its Applications, vol. 11, no. 03, Article ID 1250048, 2012.
View at: Publisher Site | Google Scholar
M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, London, 1969.
J. C. Han, “The zero-divisor graph under group actions in a non-commutative ring,” Journal of the Korean Mathematical Society, vol. 45, no. 6, pp. 1647–1659, 2008.
View at: Publisher Site | Google Scholar
Q. M. Sun, Z. L. Zhang, and J. Liu, “Soft sets and soft modules,” Lecture Notes in Computer Science, vol. 5009, pp. 403–409, 2008.
View at: Google Scholar
R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach Science Publishers, New York, NY, USA, 1991.
L. R. Vermani, An Elementary Approach to Homomlogical Algebra, Chapman and Hall, London, UK, 2003.
Q. Mushtaq and S. M. Yusuf, “On locally associative LA-semigroups,” Journal of Natural Sciences and Mathematics, vol. 1, no. 19, pp. 57–62, 1979.
View at: Google Scholar

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Copyright © 2022 Asima Razzaque and Inayatur Rehman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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