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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.13408 |
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| _version_ | 1866909464518459392 |
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| author | Tharakan, Abin Sam Sheeja, G. |
| author_facet | Tharakan, Abin Sam Sheeja, G. |
| contents | The concept of $Γ$-semigroups was introduced by M. K Sen in 1981. This study aims to investigate several intriguing properties of $Γ$-semigroups and to provide the concepts of simple $Γ$-semigroups, 0-simple $Γ$-semigroups, and completely 0-simple $Γ$-semigroups. We prove that non-zero elements of the completely 0-simple $Γ$-semigroups form a D-class and are regular. Fundamental elements of these structures are explored, and we provide concrete results that characterize them using various ideals of $Γ$-semigroups and establish the necessary and sufficient condition for a $Γ$-semigroups to be completely 0-simple. This study further introduce $Γ$-prime ideals and gave some condition in which a $Γ$-2-sided ideal to be a $Γ$-prime. In addition, we establish a condition for a commutative $Γ$ semigroup to be $Γ$-prime. we have established how union and intersection of $Γ$-prime ideals become $Γ$-prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13408 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On The Ideals of $Γ$-Semigroup Tharakan, Abin Sam Sheeja, G. Rings and Algebras The concept of $Γ$-semigroups was introduced by M. K Sen in 1981. This study aims to investigate several intriguing properties of $Γ$-semigroups and to provide the concepts of simple $Γ$-semigroups, 0-simple $Γ$-semigroups, and completely 0-simple $Γ$-semigroups. We prove that non-zero elements of the completely 0-simple $Γ$-semigroups form a D-class and are regular. Fundamental elements of these structures are explored, and we provide concrete results that characterize them using various ideals of $Γ$-semigroups and establish the necessary and sufficient condition for a $Γ$-semigroups to be completely 0-simple. This study further introduce $Γ$-prime ideals and gave some condition in which a $Γ$-2-sided ideal to be a $Γ$-prime. In addition, we establish a condition for a commutative $Γ$ semigroup to be $Γ$-prime. we have established how union and intersection of $Γ$-prime ideals become $Γ$-prime. |
| title | On The Ideals of $Γ$-Semigroup |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2501.13408 |