Structure and Spectral Theory of Non-Commutative and $n$-ary $Γ$-Semirings
Chandrasekhar Gokavarapu, D. Madhusudhana Rao
TL;DR
The paper addresses the generalization of Γ-semirings to noncommutative and $n$-ary settings, developing a unified framework for ideals, radicals, and spectra. It introduces directional ideals, $(n,m)$-type ideals, and diagonal criteria for primeness, along with a generalized radical theory and Zariski-type spectral topology. It then connects these structural notions to representation theory via $n$-ary $\Gamma$-modules and primitive ideals, culminating in a Wedderburn–Artin-type decomposition in the finite/semi-primary case. The triadic spectral geometry reveals a unifying perspective that bridges commutative and noncommutative, and binary and polyadic algebra, with computational remarks suggesting algorithmic pathways for enumeration and verification. Overall, the work provides a comprehensive algebraic-and-geometric scaffold for noncommutative and polyadic $\Gamma$-semirings and sets the stage for future computational and categorical developments.
Abstract
This paper develops the structural and spectral foundations of noncommutative and n-ary Gamma semirings, extending the commutative ternary framework established in earlier studies. We introduce left, right, and two-sided ideals in the noncommutative setting, derive quotient characterizations of prime and semiprime ideals, and construct corresponding Gamma-Jacobson radicals. For general n-ary operations, we define (n,m)-type ideals and establish diagonal criteria for n-ary primeness and semiprimeness. A unified radical theory and a Zariski-type spectral topology are then formulated, connecting primitive ideals with simple module representations. The results culminate in a noncommutative Wedderburn-Artin-type decomposition, revealing a triadic spectral geometry that unifies commutative, noncommutative, and higher-arity cases