Left localizations of left Artinian rings

TL;DR

This paper provides a detailed classification of all left localizations of left Artinian rings, showing they are finitely many, idempotent in nature, and related to the ring's simple modules.

Contribution

It explicitly describes all left denominator sets and localizations of left Artinian rings, proving finiteness and idempotent structure of these localizations.

Findings

Finitely many left localizations up to isomorphism.

Each localization is an idempotent localization.

The number of maximal left denominator sets is finite and bounded by simple modules.

Abstract

For an arbitrary left Artinian ring $R$, explicit descriptions are given of all the left denominator sets $S$ of $R$ and left localizations $S^{-1}R$ of $R$. It is proved that, up to $R$-isomorphism, there are only finitely many left localizations and each of them is an idempotent localization, i.e. $S^{-1}R\simeq S_e^{-1}R$ and ${\rm ass} (S) = {\rm ass} (S_e)$ where $S_e=\{1,e\}$ is a left denominator set of $R$ and $e$ is an idempotent. Moreover, the idempotent $e$ is unique up to a conjugation. It is proved that the number of maximal left denominator sets of $R$ is finite and does not exceed the number of isomorphism classes of simple left $R$-modules. The set of maximal left denominator sets of $R$ and the left localization radical of $R$ are described.