Left localizations of left Artinian rings

الأصل · EN

For an arbitrary left Artinian ring R, explicit descriptions are given of all the left denominator sets S of R and left localizations S⁻¹R 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⁻¹R≃ Sₑ⁻¹R and ass (S) = ass (Sₑ) where Sₑ={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.