Grading of homogeneous localization by the Grothendieck group

TL;DR

This paper generalizes a classical result in graded ring theory by showing that localizations of an M-graded ring extend naturally to G-graded rings, where G is the Grothendieck group of M.

Contribution

It introduces a new perspective on grading localization by connecting M-graded rings to G-graded rings via the Grothendieck group, broadening the classical understanding.

Findings

Localization preserves G-grading structure

Homogeneous components are characterized by fractions of homogeneous elements

Application to grading of localization rings

Abstract

The main result of this article is a fantastic generalization of a classical result in graded ring theory. In fact, our result states that if $S$ is a multiplicative set of homogeneous elements of an $M$-graded commutative ring $R=\bigoplus\limits_{m\in M}R_{m}$ with $M$ a commutative monoid, then the localization ring $S^{-1}R=\bigoplus\limits_{x\in G}(S^{-1}R)_{x}$ is a $G$-graded ring where $G$ is the Grothendieck group of $M$ and each homogeneous component $(S^{-1}R)_{x}$ is the set of all fractions $f\in S^{-1}R$ such that $f=0$ or it is of the form $f=r/s$ where $r$ is a homogeneous element of $R$ and $x=[\dg(r),\dg(s)]$. As an application, ...