Subrings of Noetherian rings

Formanek, Edward; Jategaonkar, Arun Vinayak

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

Subrings of Noetherian rings
HTML articles powered by AMS MathViewer

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.

References

Jan-Erik Björk, Noetherian and Artinian chain conditions of associative rings, Arch. Math. (Basel) 24 (1973), 366–378. MR 344286, DOI 10.1007/BF01228225
Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
Paul M. Eakin Jr., The converse to a well known theorem on Noetherian rings, Math. Ann. 177 (1968), 278–282. MR 225767, DOI 10.1007/BF01350720
David Eisenbud, Subrings of Artinian and Noetherian rings, Math. Ann. 185 (1970), 247–249. MR 262275, DOI 10.1007/BF01350264
Edward Formanek, Faithful Noetherian modules, Proc. Amer. Math. Soc. 41 (1973), 381–383. MR 379477, DOI 10.1090/S0002-9939-1973-0379477-X
Edward Formanek, Noetherian $\textrm {PI}$-rings, Comm. Algebra 1 (1974), 79–86. MR 357489, DOI 10.1080/00927877408548610
Alfred W. Goldie, The structure of Noetherian rings, Lectures on rings and modules (Tulane Univ. Ring and Operator Theory Year, 1970-1971, Vol. I), Lecture Notes in Math., Vol. 246, Springer, Berlin, 1972, pp. 213–321. MR 0393118
Robert Gordon (ed.), Ring theory, Academic Press, New York-London, 1972. MR 0330129
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
Masayoshi Nagata, A type of subrings of a noetherian ring, J. Math. Kyoto Univ. 8 (1968), 465–467. MR 236162, DOI 10.1215/kjm/1250524062
Claudio Procesi and Lance Small, Endomorphism rings of modules over $\textrm {PI}$-algebras, Math. Z. 106 (1968), 178–180. MR 233846, DOI 10.1007/BF01110128