Left localizations of left Artinian rings
نویسنده
چکیده
For an arbitrary left Artinian ring R, explicit descriptions are given of all the left denominator sets S of R and left localizations SR 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. SR ≃ S e R 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.
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