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Let S be a finite normalizing extension of a ring R. If M is an S module, then M has finite uniform dimension if and only if it has finite uniform dimension when considered as an R module. Consequently, when S is a right Goldie ring, R is also a right Goldie ring. Conversely, if R is a semiprime right Goldie ring and S is a prime ring, then S is a Goldie ring. Finally, when both S and R are semiprime right Goldie rings, the quotient ring of R embeds in the quotient ring of S. In this paper we consider a finite normalizing extension S of a ring R, and determine when the Goldie chain conditions transfer from one ring to the other. Results on chain conditions for this kind of extension have been obtained by Formanek and Jategaonkar [1], who showed that S modules are Noetherian as R modules exactly when they are Noetherian as S modules; by Lemonnier [3], who proved a similar result for the Krull dimension of S modules; and by Fisher, Lanski, and Park [2], who proved the corresponding theorem for Gabriel dimension. In the case that S is a finite centralizing extension of R, Robson and Small [4] have obtained a number of results on chain conditions, as well as other properties of such extensions. Throughout the paper, S will denote a finite normalizing extension of a ring R. Thus S = I>1 siR with s =I = IR and siR = Rsi for each i. All modules will be right unital modules. For an R submodule N of an S module M, set N' Et=I Nsi. Our first result is on finite uniform dimension and uses an idea of Lemonnier [3] to build S modules from R modules in n successive steps. This theorem was obtained independently by J. C. Robson. THEOREM 1. Any S module M has finite uniform dimension if and only if it has finite uniform dimension when considered as an R module. PROOF. Since any S submodule of M is also an R submodule, if MR has finite uniform dimension, then MS must have finite uniform dimension. For the other direction, we prove that any infinite direct sum of nonzero R submodules of M gives rise to an infinite direct sum of nonzero S submodules of M. Let {NiIi = 1, 2, . . . } be a collection of nonzero R submodules of M so that 2 Ni' is a direct sum of the R modules { Ni'), for some t < n. Consider first the possibility that (2 Ni')s-+ n q Ni = 0. In particular, Nis,,1 =# 0 for each i, and Received by the editors August 31, 1979. 1980 Mathematics Subject Classification. Primary 16A34; Secondary 16A12. ? 1980 American Mathematical Society 0002-9939/80/0000-0352/$02.25

Goldie conditions in finite normalizing extensions