Subrings of Noetherian rings

Formanek, Edward; Jategaonkar, Arun Vinayak

doi:10.1090/S0002-9939-1974-0414625-5

Subrings of Noetherian rings
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by Edward Formanek and Arun Vinayak Jategaonkar

Proc. Amer. Math. Soc. 46 (1974), 181-186

DOI: https://doi.org/10.1090/S0002-9939-1974-0414625-5

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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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