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Decomposition of Maximal orders

Decomposition of Maximal orders C. R. HAJARNAVIS AND J. C. ROBSON 1. Introduction Many classes of Noetherian rings decompose as direct products of prime and Artinian rings [2, 4, 8]. (An account and alternative proofs can also be found in [3] and [7].) Here we investigate this phenomenon for another interesting class of Noetherian rings; namely those which are maximal orders in Artinian quotient rings. It is very easy to see that if such a maximal order is indecomposable then so is its quotient ring. Our main result here is that if R is a fully bounded Noetherian (FBN) ring which is a maximal order in an Artinian ring then R is a direct product of prime rings and an Artinian ring. We give some examples to show that the extra conditions imposed here are needed. This work was accomplished while the second author was a participant in the Warwick Ring Theory Symposium. He would like to thank the staff of the Mathematics Institute, University of Warwick, for their hospitality and the United Kingdom S.E.R.C. for its financial support. The first author has obtained further results on maximal orders in Artinian rings. These will appear elsewhere. 2. Definitions, conventions and notation Let R, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/15.2.123
Publisher site
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Abstract

C. R. HAJARNAVIS AND J. C. ROBSON 1. Introduction Many classes of Noetherian rings decompose as direct products of prime and Artinian rings [2, 4, 8]. (An account and alternative proofs can also be found in [3] and [7].) Here we investigate this phenomenon for another interesting class of Noetherian rings; namely those which are maximal orders in Artinian quotient rings. It is very easy to see that if such a maximal order is indecomposable then so is its quotient ring. Our main result here is that if R is a fully bounded Noetherian (FBN) ring which is a maximal order in an Artinian ring then R is a direct product of prime rings and an Artinian ring. We give some examples to show that the extra conditions imposed here are needed. This work was accomplished while the second author was a participant in the Warwick Ring Theory Symposium. He would like to thank the staff of the Mathematics Institute, University of Warwick, for their hospitality and the United Kingdom S.E.R.C. for its financial support. The first author has obtained further results on maximal orders in Artinian rings. These will appear elsewhere. 2. Definitions, conventions and notation Let R,

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Mar 1, 1983

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