Download Exercises 5 5.1. Let A be an abelian group. Set A ∗ = HomZ(A,Q/Z

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 .
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