Algebra I: Chapters 1-3 by N. Bourbaki
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, equivalent to teams and earrings. the second one bankruptcy reviews the homes of modules and linear maps, and the 3rd bankruptcy discusses algebras, specifically tensor algebras.
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7 ALGEBRAIC STRUCTURES I (4) Let (x, y) ~ x T y be a commutative law onE; the law (X, Y) ~x T Y between subsets of E is commutative. DEFINITION 9. Let E be a magma and X a subset if E. The set if elements if E which commute with each of the elements ifX is called the centralizer ifX. Let X and Y be two subsets of E and X' and Y' their respective centralizers. IfX c Y, then Y' c X'. Let (~)lei be a family of subsets ofE and for all i E I let x; be the centralizer of~- The centralizer of lei XI is lei x;.
The order relation ~ between natural numbers is characterized by the following property: m ~ n if and only if there exists an integer p e N such that m + p = n (Set Theory, Ill,§ 3, no. 6, Proposition 13 and§ 5, no. 2, Proposition 2). The relation y - x e N between rational integers x andy is a total order relation on Z which extends the order relation ~ on N. For, for all x e Z, x - x = 0 e N; if y - x e N and z - y eN, then z - x = (z - y) + (y - x) eN, for N is stable under addition; if y - x eN and x - y eN, then y - x = 0, for 0 is the only element ofN whose negative belongs toN; for arbitrary rational integersxandy,y- x eN or x- yeN, for Z = N u ( -N); finally, ifx andy are natural numbers, then y - x eN if and only if there exists p eN such that x + p = y.
Applying Zorn's Lemma (Set Theory, III,§ 2, no. 4), we obtain the following result: PRoPosmoN 3. Let G be a group with operators, P a subset ofG and H a stable subgroup G not meeting P. The set of stable subgroups ofG containing Hand not meeting P has a maximal element. 4. QUOTIENT GROUPS Let R be an equivalence relation on a group with operators G; ifR is left (resp. right) compatible(§ 3, no. 3) with the group law on G and compatible with the action of n, the equivalence class of e is a stable subgroup H of G and the relation R is equivalent to 'x- 1y e H (resp.