Paper
Projective Modules and Classical Algebraic K-Theory of Non-Commutative Gamma Semirings
Authors
Chandrasekhar Gokavarapu
Abstract
In this paper, we initiate the study of algebraic K-theory for non-commutative -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--modules over a non-commutative -semiring . We prove that this category admits an exact structure, allowing for the definition of the Grothendieck group . Furthermore, we develop the theory of the Whitehead group using elementary matrices and the Steinberg relations in the non-commutative -semiring context. We establish the fundamental exact sequences linking and and provide explicit calculations for specific classes of non-commutative -semirings. This work lays the algebraic groundwork for future studies on higher K-theory spectra.