## Description

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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