On the number of generators of an algebra

Uriya A. First; Zinovy Reichstein

doi:10.1016/j.crma.2016.11.015

Algebra

On the number of generators of an algebra

Presented by: Jean-Pierre Serre

Uriya A. First ¹; Zinovy Reichstein ¹

¹ Department of Mathematics, University of British Columbia, Vancouver, Canada

Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 5-9.

Abstract
Résumé

A classical theorem of Forster asserts that a finite module M of rank ≤n over a Noetherian ring of Krull dimension d can be generated by $n + d$ elements. We prove a generalization of this result, with “module” replaced by “algebra”. Here we allow arbitrary finite algebras, not necessarily unital, commutative or associative. Forster's theorem can be recovered as a special case by viewing a module as an algebra where the product of any two elements is 0.

Un théorème classique de Forster affirme que tout module M de type fini et de rang ≤n sur un anneau noethérien de dimension de Krull d peut être engendré par $n + d$ éléments. Nous prouvons une généralisation de ce résultat où le mot « module » est remplacé par « algèbre ». Les algèbres considérées ici sont de type fini, mais non nécessairement unitaires, commutatives ou même associatives. Le théorème de Forster peut être déduit du cas particulier où un module est vu comme une algèbre dont le produit de deux éléments quelconques est 0.

Received: 2016-10-21
Accepted: 2016-11-29
Published online: 2016-12-07

DOI: 10.1016/j.crma.2016.11.015

Author's affiliations:

Uriya A. First ¹; Zinovy Reichstein ¹

¹ Department of Mathematics, University of British Columbia, Vancouver, Canada

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Uriya A. First; Zinovy Reichstein. On the number of generators of an algebra. Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 5-9. doi : 10.1016/j.crma.2016.11.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.11.015/

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