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

TL;DR

This paper develops a comprehensive ideal-theoretic framework for commutative ternary Γ-semirings, introducing prime and semiprime ideals, quotient criteria, radical theory, and a prime–congruence correspondence that yields a Zariski-type spectrum. It proves that a proper ideal P is prime iff T/P has no nonzero zero-divisors under the induced ternary operation, and shows semiprime ideals are closed under intersections and equal their radicals, linking to Jacobson-type radicals. The work also integrates module theory via ternary Γ-modules, proves a congruence–ideal correspondence, and defines a geometric spectrum with topological structure; these are complemented by computational classifications of small finite ternary Γ-semirings, confirming the theoretical predictions and revealing new phenomena. The results provide a rigorous algebraic and computational foundation for future categorical, geometric, and applications in coding, fuzzy logic, and higher-arity computation.

Abstract

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.

Figures (1)

• Figure 1: Ideal lattice diagram for a finite ternary $\Gamma$-semiring with two distinct prime ideals $P_{1},P_{2}$ and their meet $Q_{1}=P_{1}\cap P_{2}$.

Theorems & Definitions (161)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Example 2.4
• Example 2.5
• Definition 3.1
• Proposition 3.2: Elementary Properties
• proof
• Definition 3.3
• Theorem 3.4: Quotient Characterization of Prime Ideals