Left localizations of left Artinian rings

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