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Download Exercises 5 5.1. Let A be an abelian group. Set A ∗ = HomZ(A,Q/Z
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Transcript
Exercises 5 5.1. Let A be an abelian group. Set A∗ = HomZ (A, Q/Z). Then for any 0 6= a ∈ A, there exists some fa ∈ A∗ such that fa (a) 6= 0. Deduce that A can be embedded into a (possibly infinite) product of Q/Z. 5.2. Given a commutative diagram of abelian groups with exact rows: /A 0 / 0 f / B g / α A0 f0 / B g0 / C 0 γ / C0 / 0. (a) Show that α is a monomorphism, γ is an epimorphism, and Cokerα ∼ = Kerγ. (b) Show that there exists a commutative diagram of abelian groups with exact rows / 0 CO ∗ g∗ / B∗ f∗ γ∗ 0 / C 0∗ / / AO ∗ 0 α∗ g 0∗ / B∗ f 0∗ / A0∗ / 0. Moreover, α∗ (resp. γ ∗ ) is an isomorphism if and only if α (resp. γ) is an isomorphism. f g∗ g f∗ (c) Show that A → − B→ − C is exact if and only if C ∗ − → B ∗ −→ A∗ is exact. 5.3. Let A, B, C be modules over a commutative ring R. (a) The set L (A, B; C) of all bilinear maps A × B → C is an R-module with (f + g)(a, b) = f (a, b) + g(a, b), and (rf )(a, b) = rf (a, b). (b) Each one of the following R-modules is isomorphic to L (A, B; C): N i. HomR (A R B, C); ii. HomR (A, HomR (B, C)); iii. HomR (B, HomR (A, C)). 5.4. An algebra A over a field K is called a division algebra, if A is a division ring. Give an example of noncommutative division algebra over R. 5.5. Let K be a field, and A a K-linear space with a basis {xi }i∈I . Show that a bilinear map A × A → A, (a, b) 7→ a · b makes A an algebra (not necessarily with an identity) if and only if for any i, j, k ∈ I, (xi · xj ) · xk = xi · (xj · xk ). Deduce that any monoid M gives to a K-algebra with basis M and multiplication induced by the one in M . 1