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International Journal of Mathematics and Mathematical Sciences
Volume 2014 (2014), Article ID 315919, 13 pages
http://dx.doi.org/10.1155/2014/315919
Research Article

When an Extension of Nagata Rings Has Only Finitely Many Intermediate Rings, Each of Those Is a Nagata Ring

1Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1320, USA
2Laboratoire de Mathématiques, Université Blaise Pascal, UMR6620, CNRS, Les Cézeaux, 24 avenue des Landais, BP 80026, 63177 Aubière Cedex, France

Received 30 March 2014; Accepted 14 June 2014; Published 2 September 2014

Academic Editor: Kaiming Zhao

Copyright © 2014 David E. Dobbs et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be an extension of commutative rings, with X an indeterminate, such that the extension of Nagata rings has FIP (i.e., has only finitely many -subalgebras). Then, the number of -subalgebras of equals the number of R-subalgebras of S. In fact, the function from the set of R-subalgebras of S to the set of -subalgebras of given by is an order-isomorphism.