Localization in Associative Rings
Arvid Siqveland
TL;DR
The paper extends the notion of localization from commutative to associative rings by constructing localizing rings $A_M$ for finite sets of basepoints $M$ built from simple right $A$-modules, and by formulating a moduli-theoretic framework of schemes of associative algebras based on these basepoints. It develops both ordinary and Hausdorff localization in the associative setting, introduces local function rings (LFRs) as noncommutative analogues of local rings, and proves the existence and universal properties of $A_M$ and $H^A_M$ with respect to appropriate representation functors. It also establishes that the associative localizing rings admit categorical products, matching the commutative case where $A_M$ decomposes as a product of localizations, thereby enabling a scheme-like construction for associative algebras via basepoints. The results provide a pathway to noncommutative geometric structures by moduli of basepoints and their localizing rings, linking deformation-theoretic constructions to a noncommutative analogue of schemes.
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.