Prime and Semiprime Ideals in Commutative Ternary $\mathrm{\Gamma }$-Semirings: Quotients, Radicals, Spectrum

The theory of ternary $\mathrm{\Gamma }$-semirings extends classical ring and semiring frameworks by introducing a ternary product controlled by a parameter set $\mathrm{\Gamma }$. 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 $\mathrm{\Gamma }$-semirings and prove a quotient characterization: an ideal $P$ is prime if and only if $T\mathrm{/}P$ is free of nonzero zero-divisors under the induced ternary $\mathrm{\Gamma }$-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 $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(T)$. Computational classification of all commutative ternary $\mathrm{\Gamma }$-semirings of order $\le 4$ 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 $\mathrm{\Gamma }$-algebras.