# On rings which satisfy the minimum condition for the right-hand ideals

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## Abstract

On rings which satisfy the minimum condition for the right-hand ideals COMPOSITIO MATHEMATICA JAKOB LEVITZKI On rings which satisfy theminimum condition for the right-hand ideals Compositio Mathematica, tome 7 (1940), p. 214-222. <http://www.numdam.org/item?id=CM_1940__7__214_0> © Foundation Compositio Mathematica, 1940, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ On rings which satisfy the minimum condition for the right-hand ideals by Jakob Levitzki Jerusalem Introduction. A ring S which satisfies the double-chaim-condition (or the equivalent maximum and minimum condition) for the right- hand (in short: r.h.) ideals, possesses accoording to E. Artin 1) a nilpotent radical R, and the quotient ring S/R is semi-simple. This fact, as well as the results obtained by Artin concerning the "primary" and the "completely primary" rings attached to S are valid for a wider class of rings. In the present note it is shown that the maximum condition can be omitted without affecting the results achieved by Artin. The method used in the present note is partly an improvement of one used by the author in a previous paper 2). On the other hand, the results obtained presently yield a generalisation of the principal theorem proved in L, which can be stated now as follows: Each nil-subring of a ring which satisfies the minimum condition for the r.h. ideals, is nilpotent. This statement is in particular a solution of a problem raised by G. Kôthe 3), whether or not there exist potent nil-rings which satisfy the

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