Algebre, Chapitre 8: Modules et anneaux semi-simples - download pdf or read online

By Nicolas Bourbaki

ISBN-10: 2705612610

ISBN-13: 9782705612610

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Additional resources for Algebre, Chapitre 8: Modules et anneaux semi-simples (Elements de mathematique, Fascicule 23, ASI 1261)

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The opposite law to/). The law of right (resp. left) action associated with 8 isf (resp. the opposite law t o f ) . Let £2, E, F be sets, a ^fa an action of Q on E and a >->- ga an action of C2 on F. An 0,-morphism of E into F, or mapping of E into F compatible with the action of Q, a mapping h of E into F such that ga{h(x)) = h(fa(x)) I ALGEBRAIC STRUCTURES for all a e £2 and x e E. The composition of two £2-morphisms is an Q-morphism. Let £2, 3, E, F be sets, a an action of £2 on E, (3 i—igB an action of 3 on F and (j) a mapping of Q into £2'.

1 = x ( — I)-*= --x for x , y , z in Z. v may be written x . 1 = x . Now f x { - y ) = — f x { y ) , that is x ( — y ) = — x y ; similarly, the endomorphism y (-»• — x y of Z maps 1 to — x . whence ( — x ) . /) = -(*(-

7. APPLICATIONS: III. GENERALIZED POWERS Let E be a monoid with identity element e and law of composition denoted by T. If x is invertible in E, let g x be the homomorphism of Z into E mapping 1 to x . Let g x ( n ) = T x for all n 6 Z ; by Lemma 2 this notation is compatible for n e N with the notation introduced earlier. Then m + n / m \ /n \ T x = \T X) T \T xj (9) (10) 0 Tx—e (11) Tx= x l for x invertible in E and m , n in Z. v, the mapping n • -> gx(mn) of Z into E is a homomorphism mapping 1 t o y , whence gx(mn) = gy(n), that is (12 ) T, - T (” *).

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Algebre, Chapitre 8: Modules et anneaux semi-simples (Elements de mathematique, Fascicule 23, ASI 1261) by Nicolas Bourbaki


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