Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings
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 deca...
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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2022/2792450 |
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| author | Asima Razzaque Inayatur Rehman |
| author_facet | Asima Razzaque Inayatur Rehman |
| author_sort | Asima Razzaque |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-98e9d7f06b2b4c879a10f8f8fdb89b6d |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-98e9d7f06b2b4c879a10f8f8fdb89b6d2025-02-03T01:24:11ZengWileyJournal of Mathematics2314-47852022-01-01202210.1155/2022/2792450Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative RingsAsima Razzaque0Inayatur Rehman1Basic Science DepartmentDepartment of Mathematics and SciencesThe 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.http://dx.doi.org/10.1155/2022/2792450 |
| spellingShingle | Asima Razzaque Inayatur Rehman Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings Journal of Mathematics |
| title | Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings |
| title_full | Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings |
| title_fullStr | Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings |
| title_full_unstemmed | Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings |
| title_short | Some Developments in the Field of Homological Algebra by Defining New Class of Modules over Nonassociative Rings |
| title_sort | some developments in the field of homological algebra by defining new class of modules over nonassociative rings |
| url | http://dx.doi.org/10.1155/2022/2792450 |
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