By N. Bourbaki
This softcover reprint of the 1974 English translation of the 1st 3 chapters of Bourbaki’s Algebre offers an intensive exposition of the basics of basic, linear, and multilinear algebra. the 1st bankruptcy introduces the fundamental items, similar to teams and jewelry. the second one bankruptcy stories the homes of modules and linear maps, and the 3rd bankruptcy discusses algebras, specially tensor algebras.
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Example text
L x. The elements of n are often called operators. Examples. (1) Let E be an associative magma written multiplicatively. The mapping which associates with a strictly positive integer n the mapping x >-+ xn of E into itself is an action of N* on E. IfE is a group, the mapping which associates with a rational integer a the mapping x >-+ xa of E into E is an action of Z on E. (2) Let E be a magma with law denoted by T. The mapping which associates with x e E the mapping A >-+ x T A of the set of subsets of E into itself is an action ofE on t;p(E).
1) express the fact thatfis a homomorphism ofN into E and obviously f(l) = x. Ifj' is a homomorphism of N into E such thatf'(l) = x, then f = j', by§ 1, no. 4, Proposition 1, (iv). Suppose now that x is invertible. By no. 3, Corollary 2 to Proposition 4, . : 0. By construction, Z is the group of differences ofN and hence (no. 4, Theorem l)fextends uniquely to a homomorphism g of Z into E. ' of g' toN is a homomorphism of N into E withf'(1) = x. Hencef' =f. whenceg' =g. We shall apply Lemma 2 to the case where the monoid E is Z; for every integer m E Z there therefore exists an endomorphismfm of Z characterized by fm( 1) = m.
Further, ify = T x, the mapping n ~ gx(mn) ofZ into Eisa homomorphism mapping I toy, whence gx(mn) = g 11 (n), that is mn " (m ) TX=T Tx. (12) -1 1 As -1 is the negative of I in Z, T xis the inverse of x = T x in E. If we write n -m m = -min (9), it is seen that T xis the inverse ofT x. 8. NOTATION (a) As a general rule the law of a commutative monoid is written additively. It is then a convention that -x denotes the negative of x. The notation x + (-y) is abbreviated to x - y and similarly X + y - z, X - y - z, X- y +Z- t, etc....