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

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.

💡 Research Summary

The paper develops the first systematic ideal theory for commutative ternary Γ‑semirings, a recent generalization of rings and semirings where the binary product is replaced by a ternary operation controlled by an external parameter set Γ. After recalling the axioms introduced by Rao, Rani and Kiran (2025), the authors define ideals in the natural way: a non‑empty subset I of T is an ideal if it is a sub‑semigroup under addition and whenever any one of the three entries of a ternary product a α b β c lies in I, the whole product lies in I.

A prime ideal P is defined by the condition that a α b β c ∈ P forces at least one of a, b, c to belong to P. The authors prove elementary properties (e.g., the product of three ideals contained in P forces one of them into P) and show that the intersection of finitely many prime ideals need not be prime, providing explicit counter‑examples with the semiring of natural numbers under the ternary sum operation. The central quotient characterization (Theorem 3.4) states that P is prime if and only if the quotient ternary Γ‑semiring T/P has no non‑zero zero‑divisors, where a zero‑divisor is an element x for which there exist non‑zero y, z and parameters α, β with x α y β z = 0. This result mirrors the classical binary case but requires a careful adaptation to the three‑operand setting.

The paper then introduces maximal ideals, proves that every maximal ideal is prime, and shows that quotients by maximal ideals are simple ternary Γ‑semirings (they have no non‑trivial ideals). Primary ideals are defined by a weakened condition: if a α b β c ∈ Q and a ∉ Q, then either b α b β b ∈ Q or c α c β c ∈ Q. Every prime ideal is shown to be primary, while examples demonstrate primary ideals that are not prime.

A major contribution is the study of semiprime ideals. The authors prove that the class of semiprime ideals is closed under arbitrary intersections and that each semiprime ideal coincides with its own radical (the smallest semiprime ideal containing it). Consequently, the radical of any ideal can be described as the intersection of all prime ideals containing it, establishing a direct analogue of the classical radical theory.

The paper establishes a bijection between prime ideals and prime congruences: for any ideal I, the relation a ≡ b (mod I) defines a congruence, and the map I ↦ ρ_I gives a one‑to‑one correspondence between prime ideals and prime congruences. Using this correspondence, the authors define the spectrum Spec(T) as the set of all prime ideals equipped with a Zariski‑like topology whose closed sets are V(I) = {P ∈ Spec(T) | I ⊆ P}. They verify that this topology satisfies the usual axioms (arbitrary intersections, finite unions) and discuss basic properties such as compactness in the finite case.

A substantial computational component classifies all commutative ternary Γ‑semirings of order ≤ 4. The authors implement an enumeration algorithm that generates all possible underlying sets, parameter sets Γ, and ternary operations satisfying the axioms, then computes the ideal lattice for each structure. The classification confirms theoretical predictions: (i) prime ideals are more abundant than maximal ideals, (ii) semiprime ideals are indeed closed under intersections, and (iii) novel phenomena appear, such as three distinct elements whose ternary product is zero while none of them is zero—a situation impossible in binary semiring theory. Tables of the classified structures, together with explicit examples of prime, primary, maximal, and semiprime ideals, are provided.

In the concluding section the authors outline future directions: extending the theory to non‑commutative ternary Γ‑semirings, developing a module theory over such semirings, investigating categorical frameworks, and applying the spectral construction to fuzzy logic and multi‑valued algebraic systems. Overall, the paper furnishes a rigorous algebraic foundation for ternary Γ‑semirings, bridging ideal theory, radical theory, and spectral topology, and opens a pathway for geometric and computational explorations in this emerging algebraic landscape.