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 normal, linear, and multilinear algebra. the 1st bankruptcy introduces the fundamental gadgets, reminiscent of teams and earrings. the second one bankruptcy experiences the homes of modules and linear maps, and the 3rd bankruptcy discusses algebras, in particular tensor algebras.

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Additional info for Algebra I: Chapters 1-3

Example text

Similarly we write T x = e for arbitrary x. With these definitions Theorems 1 and 3 of§ 1 remain true if the hypothesis that the sets A and B1 are ;:n x = ( Tx) T ( Tx) non-empty is suppressed. Similarly the formulae m. and T x = T(Tx) are then true form ~ 0, n ~ 0 . Let E be a unital magma whose law is denoted by T and e its identity element . tt oft e. Let (x1) 1 e 1 be a family of elements of E with finite support. We shall define the composition T x1 in the two following cases: lei (a) the set I is totally ordered; (b) E is associative and the x1 are pairwise permutable.

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).

5 n eN; it admits as negative in Z the class of elements (n, m + n). Every element (p, q) ofN x N may be written in the form (m + n, n) ifp ~ q or in the form (n, m + n) if p ~ q; it follows that Z is the union ofN and the set of negatives of the elements of N. The identity element 0 is the only element of N whose negative belongs toN. For every natural number m, - m denotes the negative rational integer of m and -N denotes the set of elements -m forme N. Then Z = N u (-N) and N n ( -N) = {0}. for m e N, m = - m if and only if m = 0.