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 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.
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 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.
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@article{CRMATH_2017__355_1_5_0, author = {First, Uriya A. and Reichstein, Zinovy}, title = {On the number of generators of an algebra}, journal = {Comptes Rendus. Math\'ematique}, pages = {5--9}, publisher = {Elsevier}, volume = {355}, number = {1}, year = {2017}, doi = {10.1016/j.crma.2016.11.015}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2016.11.015/} }
TY - JOUR AU - First, Uriya A. AU - Reichstein, Zinovy TI - On the number of generators of an algebra JO - Comptes Rendus. Mathématique PY - 2017 SP - 5 EP - 9 VL - 355 IS - 1 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2016.11.015/ DO - 10.1016/j.crma.2016.11.015 LA - en ID - CRMATH_2017__355_1_5_0 ER -
%0 Journal Article %A First, Uriya A. %A Reichstein, Zinovy %T On the number of generators of an algebra %J Comptes Rendus. Mathématique %D 2017 %P 5-9 %V 355 %N 1 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2016.11.015/ %R 10.1016/j.crma.2016.11.015 %G en %F CRMATH_2017__355_1_5_0
First, Uriya A.; Reichstein, Zinovy. On the number of generators of an algebra. Comptes Rendus. Mathématique, Tome 355 (2017) no. 1, pp. 5-9. doi : 10.1016/j.crma.2016.11.015. http://archive.numdam.org/articles/10.1016/j.crma.2016.11.015/
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