Algebra II - Noncommunicative Rings, Identities - download pdf or read online

By A. I. Kostrikin, I. R. Shafarevich

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Sec. 3 Some Set Theory 31 Definition. A function f : X → Y has an inverse if there is a function g : Y → X with both composites g ◦ f and f ◦ g being identity functions. 47. (i) If f : X → Y and g : Y → X are functions such that g ◦ f = 1 X , then f is injective and g is surjective. (ii) A function f : X → Y has an inverse g : Y → X if and only if f is a bijection. Proof. (i) Suppose that f (x) = f (x ); apply g to obtain g( f (x)) = g( f (x )); that is, x = x [because g( f (x)) = x], and so f is injective.

Note that zz = a 2 + b2 = |z|2 , so that z = 0 if and only if zz = 0. If z = 0, then z −1 = 1/z = z/zz = (a/zz) − i(b/zz); that is, 1 = a + ib a2 a + b2 −i a2 b + b2 4 If this is not clear, look at the proof of the division algorithm on page 131. Sec. 2 Roots of Unity 23 If |z| = 1, then z −1 = z. In particular, if z is a root of unity, then its reciprocal is its complex conjugate. Complex conjugation satisfies the following identities: z + w = z + w; zw = z w; z = z; z = z if and only if z is real.

Conversely, if [x] = [y], then x ∈ [x], by reflexivity, and so x ∈ [x] = [y]. Therefore, x ≡ y. • Definition. A family of subsets Ai of a set X is called pairwise disjoint if Ai ∩ A j = ∅ for all i = j. A partition of a set X is a family of pairwise disjoint nonempty subsets, called blocks, whose union is all of X . 54. If ≡ is an equivalence relation on a set X , then the equivalence classes form a partition of X . Conversely, given a partition {Ai : i ∈ I } of X , there is an equivalence relation on X whose equivalence classes are the blocks Ai .

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Algebra II - Noncommunicative Rings, Identities by A. I. Kostrikin, I. R. Shafarevich


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