Structure and Spectral Theory of Non-Commutative and $n$-ary $\Gamma$-Semirings

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