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Normal pairs of noncommutative rings

  • David E. Dobbs
  • Noômen JarbouiEmail author
Article

Abstract

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if \(R \subseteq S\) are rings and \(D=(d_{ij})\) is an \(n\times n\) matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of \(n\times n\) matrices with entries in R if and only if each \(d_{ij}\) is integral over R. If \(R \subseteq S\) are rings with corresponding full rings of \(n\times n\) matrices \(R_n\) and \(S_n\), then \((R_n,S_n)\) is a normal pair if and only if (RS) is a normal pair. Examples are given of a pair \((\Lambda , \Gamma )\) of noncommutative (in fact, full matrix) rings such that \(\Lambda \subset \Gamma \) is (resp., is not) a minimal ring extension; it can be further arranged that \((\Lambda , \Gamma )\) is a normal pair or that \(\Lambda \subset \Gamma \) is an integral extension.

Keywords

Associative ring Integrality Ring extension Normal pair Matrix Full matrix ring Minimal ring extension Prüfer domain Idealization 

Mathematics Subject Classification

Primary 16999 13B21 Secondary 13B99 13F05 13C15 

Notes

Acknowledgements

The authors wish to thank the referee for his/her very precise critique.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TennesseeKnoxvilleUSA
  2. 2.Département de Mathématiques, Faculté des Sciences de SfaxUniversité de SfaxSfaxTunisia

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