Localization in Associative Rings

TL;DR

This paper extends the concept of schemes to associative rings by constructing localizations and basepoints, generalizing the classical scheme theory from commutative to non-commutative rings.

Contribution

It introduces a set of basepoints for associative rings and proves the existence of localizing rings, enabling a scheme-like framework for non-commutative algebra.

Findings

Existence of localizing rings for finite subsets of basepoints.

Construction of a topology on basepoints that generalizes the Zariski topology.

Framework aligns with classical schemes in the commutative case.

Abstract

In arXiv:2511.04191 we constructed schemes of objects in small categories which contained a set of basepoints with local representing (localizing) objects. Here we prove that the category $\cat{Rings}$ of associative rings with unit has a certain set of basepoints for which localizing rings exist. We take the set of base points $B$ to be the set of rings on the form $\enm_{\mathbb Z}(M)$ where $M$ is a simple right $A$-module for some associative ring $A.$ The set of base-points in the associative ring $A$ is defined as $\pts_B(A)=\{\mor_{\cat{Rings}}(A,\enm_{\mathbb Z}(M))\}.$ For any finite subset $M\subseteq\pts_B(A)$ we prove that the localizing ring $A_M$ exists. and so the construction from arXiv:2511.04191 gives a definition of schemes of associative algebras. Defining a topology on $\pts_B(A)$ such that when $A$ is commutative it is the Zariski topology, we get the ordinary definition of schemes when we consider the category of commutative rings. This article is in line with the philosophy that a scheme is a moduli of its base-points.