Algebra
Generating regular elements
Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 429-432.

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.

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.

Accepted:
Published online:
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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