By Antoine Chambert-Loir

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Extra info for Algèbre commutative [Lecture notes]

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Soit en effet (Ji ) une famille totalement ordonnée d’idéaux à gauche de A tels que I ⊂ Ji A ; soit J la réunion des idéaux Ji (si la famille n’est pas vide, J = I sinon). Montrons que l’on a J ∈ I . Remarquons que J contient I par construction ; comme 1 n’appartient à Ji pour aucun indice i , 1 ∈ J et J = A. Enfin, J est un idéal à gauche de A : si x ∈ J et y ∈ J, il existe des indices i et j tels que x ∈ Ji et y ∈ J j . Commme la famille (Ji ) est totalement ordonnée, on a Ji ⊂ J j ou J j ⊂ Ji .

Alors, u(bs − at ) = 0, d’où (b, t ) ∼ (a, s). La relation est symétrique ; 28 CHAPITRE 1. ANNEAUX, IDÉAUX, ALGÈBRES – enfin, si (a, s) ∼ (b, t ) et (b, t ) ∼ (c, u), choisissons v et w ∈ S tels que v(at − bs) = w(bu − ct ) = 0. Comme t (au − cs) = u(at − bs) + s(bu − ct ), on a vwt (au − cs) = 0. Puisque v, w et t appartiennent à S, vwt ∈ S et (a, s) ∼ (c, u). La relation est donc transitive. On désigne par S −1 A l’ensemble des classes d’équivalence (on trouve parfois la notation AS ) ; la classe du couple (a, s) est notée a/s.

2) Soit A un anneau et soit M un A-module à droite. a) Montrer que l’ensemble (0 : M) des a ∈ A tels que ma = 0 pour tout m ∈ M est un idéal bilatère de A (annulateur de M). On le note aussi ann(M). b) Plus généralement, soit N un sous-A-module de M. Montrer que l’ensemble (N : M) des a ∈ A tels que ma ∈ N pour tout m ∈ M est un idéal bilatère de A. 48 CHAPITRE 2. MODULES 3) Soit A et B deux anneaux et f : A −→ B un homomorphisme d’anneaux. a) Soit M un B-module à droite. Montrer que l’on définit un A-module à droite en munissant le groupe abélien M de la multiplication (M, B) → M définie par (m, b) → m f (b).

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Algèbre commutative [Lecture notes] by Antoine Chambert-Loir


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