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Math 512 Problem Set 3

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Math 512 Problem Set 3
Exercise 1. Let R be a ring, and M an abelian group. Define
HomZ(R, M) = {f : R → M | f is a Z-module homomorphism} .
Show that HomZ(R, M) is an R-module with multiplication (rf)(x) = rf(x)
for any r ∈ R, f ∈ HomZ(R, M), and x ∈ R.
Exercise 2. Show that Q is not a projective Z-module.
Exercise 3. Show that every projective abelian group is free.
Exercise 4. Show that a direct product of R-modules Q
i∈I
Ji
is injective
if and only if each Ji
is injective.
Exercise 5. Let R be a commutative, unital ring. Show that the following
are equivalent.
(i) Every R-module is projective.
(ii) Every R-module is injective.
(iii) Every short exact sequence of R-modules is split exact.
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