Comptes Rendus
no. 11-12
Algebra
Generating regular elements
[Engendrer des éléments réguliers]

Présenté par : Michèle Vergne

J.T. Stafford1
1 School of Mathematics, Alan Turing Building, The University of Manchester, Oxford Road, Manchester M13 9PL, England, United Kingdom
Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 429-432.

Soit R un anneau de Goldie premier. Un résultat utile est que si a,b∈R sont tels que, aR+bR contienne un élément régulier, alors il existe λ∈R tel que a+bλ est régulier. Nous montrons quʼun résultat analogue est vrai pour n⩾1 paires de tels élément : si R contient un corps de cardinal >n et si les ai,bi∈R sont tels que aiR+biR contienne un élément régulier, alors il existe λ∈R tel que ai+biλ est régulier pour tout i.

Let R be a prime right Goldie ring. A useful fact is that, if a,b∈R are such that aR+bR contains a regular element, then there exists λ∈R such that a+bλ is regular. We show that the analogous result holds for n⩾1 pairs of elements: if R contains a field of cardinality at least n+1, and if ai,bi∈R are such that aiR+biR contains a regular element for 1⩽i⩽n, then there exists a single element λ∈R such that ai+biλ is regular for each i.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.06.001
J.T. Stafford 1

1 School of Mathematics, Alan Turing Building, The University of Manchester, Oxford Road, Manchester M13 9PL, England, United Kingdom 
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J.T. Stafford. Generating regular elements. Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 429-432. doi : 10.1016/j.crma.2013.06.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.06.001/

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[3] S. Carpentier; A. De Sole; V.G. Kac Rational matrix pseudodifferential operators, 2012 (preprint) | arXiv

[4] A. De Sole; V.G. Kac Non-local Poisson structures and applications to the theory of integrable systems, 2013 (preprint) | arXiv

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[6] L.W. Small; J.T. Stafford Regularity of zero divisors, Proc. Lond. Math. Soc., Volume 44 (1982) no. 3, pp. 405-419

[7] J.T. Stafford Stable structure of noncommutative Noetherian rings, J. Algebra, Volume 47 (1977) no. 2, pp. 244-267

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