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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Subrings of Noetherian rings


Authors: Edward Formanek and Arun Vinayak Jategaonkar
Journal: Proc. Amer. Math. Soc. 46 (1974), 181-186
MSC: Primary 16A46
MathSciNet review: 0414625
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0414625-5
PII: S 0002-9939(1974)0414625-5
Keywords: Commutative Noetherian rings, noncommutative Noetherian rings, P. I. rings, Eakin's theorem, Clifford's theorem on group algebras
Article copyright: © Copyright 1974 American Mathematical Society