Generating regular elements

J.T. Stafford

doi:10.1016/j.crma.2013.06.001

Algebra

Generating regular elements

Presented by: Michèle Vergne

J.T. Stafford ¹

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.

Abstract
Résumé

Let R be a prime right Goldie ring. A useful fact is that, if $a, b \in R$ are such that $a R + b R$ contains a regular element, then there exists $λ \in R$ such that $a + b λ$ 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 $a_{i}, b_{i} \in R$ are such that $a_{i} R + b_{i} R$ contains a regular element for $1 ⩽ i ⩽ n$ , then there exists a single element $λ \in R$ such that $a_{i} + b_{i} λ$ is regular for each i.

Soit R un anneau de Goldie premier. Un résultat utile est que si $a, b \in R$ sont tels que, $a R + b R$ contienne un élément régulier, alors il existe $λ \in R$ tel que $a + b λ$ 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 $a_{i}, b_{i} \in R$ sont tels que $a_{i} R + b_{i} R$ contienne un élément régulier, alors il existe $λ \in R$ tel que $a_{i} + b_{i} λ$ est régulier pour tout i.

Received: 2013-05-24
Accepted: 2013-06-11
Published online: 2013-06-01

DOI: 10.1016/j.crma.2013.06.001

Author's affiliations:

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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     title = {Generating regular elements},
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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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Cited by

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