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New examples of maximal orders. (English) Zbl 0663.16003

Let \(R\) be a left and right Noetherian ring. It is shown that \(R\) is a maximal order in a simple Artinian ring if \(R\) has an invertible ideal \(P\) such that \(P\) is contained in the Jacobson radical of \(R\) and \(R/P\) is a maximal order in a simple Artinian ring. This is used to show that if \(R\) is a maximal order in a simple Artinian ring then so also is the ring of power series over \(R\) twisted by an automorphism of \(R\). Examples are given to show that the condition that \(P\) is contained in the Jacobson radical can not be dropped, and that \(R/P\) being semi-simple Artinian is not enough.

MSC:

16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16P40 Noetherian rings and modules (associative rings and algebras)
16Kxx Division rings and semisimple Artin rings
16N60 Prime and semiprime associative rings
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
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References:

[1] Ghamarie M., Vorlesungen aus dem Fachbereich Mathematik der Universitat Essen Heft 3 (1979)
[2] DOI: 10.1016/0021-8693(72)90105-6 · Zbl 0241.16002 · doi:10.1016/0021-8693(72)90105-6
[3] Chatters A. W., Rings with chain conditions (1980) · Zbl 0446.16001
[4] DOI: 10.1112/jlms/s2-9.2.337 · Zbl 0294.16019 · doi:10.1112/jlms/s2-9.2.337
[5] DOI: 10.1016/0021-8693(72)90007-5 · Zbl 0241.16003 · doi:10.1016/0021-8693(72)90007-5
[6] DOI: 10.1080/00927878608823381 · Zbl 0601.16005 · doi:10.1080/00927878608823381
[7] DOI: 10.1007/BFb0090774 · doi:10.1007/BFb0090774
[8] DOI: 10.1016/0021-8693(68)90025-2 · Zbl 0164.03904 · doi:10.1016/0021-8693(68)90025-2
[9] DOI: 10.1112/plms/s3-22.1.39 · Zbl 0208.29703 · doi:10.1112/plms/s3-22.1.39
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