Left localizations of left Artinian rings

Abstract

For an arbitrary left Artinian ring [Formula: see text], explicit descriptions are given of all the left denominator sets [Formula: see text] of [Formula: see text] and left localizations [Formula: see text] of [Formula: see text]. It is proved that, up to [Formula: see text]-isomorphism, there are only finitely many left localizations and each of them is an idempotent localization, i.e. [Formula: see text] and [Formula: see text] where [Formula: see text] is a left denominator set of [Formula: see text] and [Formula: see text] is an idempotent. Moreover, the idempotent [Formula: see text] is unique up to a conjugation. It is proved that the number of maximal left denominator sets of [Formula: see text] is finite and does not exceed the number of isomorphism classes of simple left [Formula: see text]-modules. The set of maximal left denominator sets of [Formula: see text] and the left localization radical of [Formula: see text] are described.