Projective Modules and Classical Algebraic K-Theory of Non-Commutative Gamma Semirings

TL;DR

This paper develops the algebraic K-theory for non-commutative Gamma-semirings, extending classical concepts like Grothendieck and Bass K-theory to this new setting, including foundational categorical and algebraic structures.

Contribution

It introduces the category of finitely generated projective bi-Gamma-modules over non-commutative Gamma-semirings and constructs their K-theory groups, laying groundwork for higher K-theory.

Findings

Defined the category of finitely generated projective bi-Gamma-modules

Constructed the Grothendieck group K_0^Gamma(T) for non-commutative Gamma-semirings

Established fundamental exact sequences linking K_0 and K_1

Abstract

In this paper, we initiate the study of algebraic K-theory for non-commutative $\Gamma$-semirings, extending the classical constructions of Grothendieck and Bass to this setting. We first establish the categorical foundations by constructing the category of finitely generated projective bi-$\Gamma$-modules over a non-commutative $\Gamma$-semiring $T$. We prove that this category admits an exact structure, allowing for the definition of the Grothendieck group $K_0^\Gamma(T)$. Furthermore, we develop the theory of the Whitehead group $K_1^\Gamma(T)$ using elementary matrices and the Steinberg relations in the non-commutative $\Gamma$-semiring context. We establish the fundamental exact sequences linking $K_0$ and $K_1$ and provide explicit calculations for specific classes of non-commutative $\Gamma$-semirings. This work lays the algebraic groundwork for future studies on higher K-theory spectra.