Prime and Semiprime Ideals in Commutative Ternary $Γ$-Semirings: Quotients, Radicals, Spectrum
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| Format: | Preprint |
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2025
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| author | Gokavarapu, Chandrasekhar Rao, Dr D Madhusudhana |
| author_facet | Gokavarapu, Chandrasekhar Rao, Dr D Madhusudhana |
| contents | The theory of ternary $Γ$-semirings extends classical ring and semiring frameworks by introducing a ternary product controlled by a parameter set $Γ$. Building on the foundational axioms recently established by Rao, Rani, and Kiran (2025), this paper develops the first systematic ideal-theoretic study within this setting. We define and characterize prime and semiprime ideals for commutative ternary $Γ$-semirings and prove a quotient characterization: an ideal $P$ is prime if and only if $T/P$ is free of nonzero zero-divisors under the induced ternary $Γ$-operation. Semiprime ideals are shown to be stable under arbitrary intersections and coincide with their radicals, providing a natural bridge to radical and Jacobson-type structures. A correspondence between prime ideals and prime congruences is established, leading to a Zariski-like spectral topology on $\mathrm{Spec}(T)$. Computational classification of all commutative ternary $Γ$-semirings of order $\leq 4$ confirms the theoretical predictions and reveals novel structural phenomena absent in binary semiring theory. The results lay a rigorous algebraic and computational foundation for subsequent categorical, geometric, and fuzzy extensions of ternary $Γ$-algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_23885 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prime and Semiprime Ideals in Commutative Ternary $Γ$-Semirings: Quotients, Radicals, Spectrum Gokavarapu, Chandrasekhar Rao, Dr D Madhusudhana Rings and Algebras Primary 16Y60, 16Y90, Secondary 08A30, 06B10, 16N60, 68W30 The theory of ternary $Γ$-semirings extends classical ring and semiring frameworks by introducing a ternary product controlled by a parameter set $Γ$. Building on the foundational axioms recently established by Rao, Rani, and Kiran (2025), this paper develops the first systematic ideal-theoretic study within this setting. We define and characterize prime and semiprime ideals for commutative ternary $Γ$-semirings and prove a quotient characterization: an ideal $P$ is prime if and only if $T/P$ is free of nonzero zero-divisors under the induced ternary $Γ$-operation. Semiprime ideals are shown to be stable under arbitrary intersections and coincide with their radicals, providing a natural bridge to radical and Jacobson-type structures. A correspondence between prime ideals and prime congruences is established, leading to a Zariski-like spectral topology on $\mathrm{Spec}(T)$. Computational classification of all commutative ternary $Γ$-semirings of order $\leq 4$ confirms the theoretical predictions and reveals novel structural phenomena absent in binary semiring theory. The results lay a rigorous algebraic and computational foundation for subsequent categorical, geometric, and fuzzy extensions of ternary $Γ$-algebras. |
| title | Prime and Semiprime Ideals in Commutative Ternary $Γ$-Semirings: Quotients, Radicals, Spectrum |
| topic | Rings and Algebras Primary 16Y60, 16Y90, Secondary 08A30, 06B10, 16N60, 68W30 |
| url | https://arxiv.org/abs/2510.23885 |