By N. Bourbaki

This softcover reprint of the 1974 English translation of the 1st 3 chapters of Bourbaki’s Algebre provides an intensive exposition of the basics of basic, linear, and multilinear algebra. the 1st bankruptcy introduces the elemental items, similar to teams and earrings. the second one bankruptcy experiences the houses of modules and linear maps, and the 3rd bankruptcy discusses algebras, specially tensor algebras.

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**Example text**

P*) by (1), hence e is surjective. I. Let E be a commutative monoid, S a subset rifE, E 8 the monoid riffractions associated with S and e: E ~ E8 the canonical homomorphism. Further let f be a homomorphism rifE into a monoid F (not necessarily commutative) such that every element off(S) is invertible in F. There exists one and only one homomorphism J rif E 8 into F such that f = J o e. f(p)*. We show that g is a homomorphism of E x S' into F. First of all, g(e, e) = f(e)f(e)* = e. Let (a,p) and (a',p') be two elements ofE x S'; as a' andp commute in E, f(a') andf(p) commute in F, whencef(a')f(p)* = f(p)*f(a') by no.

The canonical mapping ofEJT onto (EJR)f(TJR) (Set Theory, II, § 6, no. 7) is then a magma isomorphism. PROPOSITION 8. Let E be a magma, A a stable subset of E and R an equivalence relation onE compatible with the law on E. The saturation B of A with respect toR (Set 11 ALGEBRAIC STRUCTURES Theory, II,§ 6, no. 5) is a stable subset. The equivalence relations RA and R 8 induced by R on A and B respectively are compatible with the induced laws and the mapping derived from the canonical injection of A into B by passing to the quotients is a magma isomorphism of AfRA onto BfR8 .

CoROLLARY. Let E be an associative magma. The centralizer if any subset if E is a stable subset if E. DEFINITION 10. The centralizer if a magma E is called the centre if E. An element of the centre if E is called a central element if E. If E is an associative magma its centre is a stable subset by the Corollary to Proposition 3 and the law induced on its centre is commutative. PRoPOSITION 4. Let E be an associative magma, X and Y two subsets if E. lf every element if X commutes with every element ifY every element if lhe stable subset generated by X commutes with every element of the stable subset generated by Y.