Grading of homogeneous localization by the Grothendieck group

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 $R=\underset{m\in M}{⨁}{R}_m$ with $M$ a commutative monoid, then the localization ring $S^{-1}R=\underset{x\in G}{⨁}(S^{-1}R{)}_x$ 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, ...

10 pages