Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1974-0414625-5
MathSciNet review: 0414625
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] J. E. Björk, On Noetherian and Artinian chain conditions of associative rings, Arch. Math. 24 (1973), 366-378. MR 0344286 (49:9025)
  • [2] C. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Appl. Math., vol. 11, Interscience, New York, 1962. MR 26 #2519. MR 0144979 (26:2519)
  • [3] P. M. Eakin, Jr., The converse to a well-known theorem on Noetherian rings, Math. Ann. 177 (1968), 278-282. MR 37 #1360. MR 0225767 (37:1360)
  • [4] D. Eisenbud, Subrings of Artinian and Noetherian rings, Math. Ann. 185 (1970), 247-249. MR 41 #6885. MR 0262275 (41:6885)
  • [5] E. Formanek, Faithful Noetherian modules, Proc. Amer. Math. Soc. 41 (1973), 381-383. MR 0379477 (52:382)
  • [6] -, Noetherian P. I. rings, Communications in Algebra 1 (1974), 79-86. MR 0357489 (50:9957)
  • [7] A. W. Goldie, The structure of Noetherian rings, Lectures on Rings and Modules, Lecture Notes in Math., vol. 246, Springer-Verlag, New York, 1972. MR 0393118 (52:13928)
  • [8] R. Gordon, Ring theory, Academic Press, New York, 1972. MR 0330129 (48:8467)
  • [9] I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7234. MR 0254021 (40:7234)
  • [10] M. Nagata, A type of subrings of a noetherian ring, J. Math. Kyoto Univ. 8 (1968), 465-467. MR 38 #4460. MR 0236162 (38:4460)
  • [11] C. Procesi and L. W. Small, Endomorphism rings of modules over PI-algebras, Math. Z. 106 (1968), 178-180. MR 38 #2167. MR 0233846 (38:2167)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 16A46

Retrieve articles in all journals with MSC: 16A46


Additional Information

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

American Mathematical Society