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

TL;DR

This paper introduces a systematic ideal-theoretic framework for commutative ternary $ ext{Gamma}$-semirings, defining prime and semiprime ideals, establishing their properties, and exploring their spectrum and computational classification.

Contribution

It is the first to develop an ideal theory for commutative ternary $ ext{Gamma}$-semirings, including prime and semiprime ideals, quotient characterizations, and spectral topology.

Findings

Prime ideals characterized by zero-divisor-free quotients

Semiprime ideals stable under intersections and equal to radicals

Classification of small-order ternary $ ext{Gamma}$-semirings confirms theoretical results

Abstract

The theory of ternary $\Gamma$-semirings extends classical ring and semiring frameworks by introducing a ternary product controlled by a parameter set $\Gamma$. 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 $\Gamma$-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 $\Gamma$-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 $\mathrm{Spec}(T)$. Computational classification of all commutative ternary $\Gamma$-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 $\Gamma$-algebras.