Left localizations of left Artinian rings

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 $\mathrm{ass}(S)=\mathrm{ass}(S_e)$ 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.

31 pages

References in corpus (2)

• Left localizable rings and their characterizations
• Characterizations of left orders in left Artinian rings

Cited by in corpus (1)

• New criteria for a ring to have a semisimple left quotient ring