Generating regular elements

J.T. Stafford

doi:10.1016/j.crma.2013.06.001

Algebra

Generating regular elements

Michèle Vergne

J.T. Stafford ¹

¹School 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 (Original language)
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.

Article information
Cite this article

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

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

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