PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 2, November 1974 SUBRINGS OF NOETHERIAN RINGS EDWARD FORMANEK AND ARUN VINAYAK JATEGAONKAR 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 Rç has a finite generating set {u ,,•••, u ! such that u -S = Su. for 1 < í < m, then a right R-module is Noetherian, Artinian or semi- simple iff it is respectively so as a right S-module. This yields a re suit of Clifford on group algebras. Let S be a subring of a ring R such that R is finitely generated as a right S-module. It is well known (and trivial) that if S is a right Noe- therian ring then R is a right Noetherian ring. The converse is false in gen- eral as can be seen by taking R = (q q) and S = ( j ^). However, P. M. Eakin [3] (and later M. Nagata [10]) showed that the converse holds if R is assumed to be a commutative ring. D. Eisenbud [4] and J. E. Björk [1] have extended Eakin's theorem to some mildly noncommutative situa- tions. In this note, we provide two mildly noncommutative versions of Eakin's theorem. The first version answers a question raised by J. E. Björk [8J. The second version improves upon Eisenbud's generalization of Eakin's theorem and, when applied to group algebras, yields a theorem of Clifford [2, p. 343]. As usual, all rings, subrings and modules are assumed to be unitary. Recall that a module MR is finite dimensional if it does not contain any infinite direct sum of nonzero submodules. It is known [7, p. 216] that if Mr, is finite dimensional then there exists a nonnegative integer n such that any direct sum of nonzero submodules of M contains at most 22 terms. The least such integer is called the uniform dimension of M and is denoted as d(M) or dAM). _ K Received by the editors September 17, 1973. AMS(MOS) subject classifications(l970). Primary 16A46, 13E05; Secondary 16A26, 16A38. Key words and phrases. Commutative Noetherian rings, noncommutative Noe- therian rings, P. I. rings, Eakin's theorem, Clifford's theorem on group algebras. Copyright © 1974, American Mathematical Society 181 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use