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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Subrings of Noetherian rings
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by Edward Formanek and Arun Vinayak Jategaonkar PDF
Proc. Amer. Math. Soc. 46 (1974), 181-186 Request permission

Abstract:

Let $S$ be a subring of a ring $R$ such that $R$ is a finitely generated right $S$-module. Clearly, if $S$ is a right Noetherian ring then so is $R$. Generalizing a result of P. M. Eakin, we show that if $R$ is right Noetherian and $S$ is commutative then $S$ is Noetherian. We also show that if ${R_S}$ has a finite generating set $\{ {u_1}, \cdots ,{u_m}\}$ such that ${u_i}S = S{u_i}$ for $1 \leq i \leq m$, then a right $R$-module is Noetherian, Artinian or semisimple iff it is respectively so as a right $S$-module. This yields a result of Clifford on group algebras.
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 46 (1974), 181-186
  • MSC: Primary 16A46
  • DOI: https://doi.org/10.1090/S0002-9939-1974-0414625-5
  • MathSciNet review: 0414625