Exercises 3
3.1. Let R be a ring with identity. An R-module A is injective if and only if for every left
ideal L of R and R-module homomorphism g : L → A, there exists a ∈ A such that
g(r) = ra for every r ∈ L.
3.2. Every vector space over a division ring D is both a projective and an injective Dmodule.
3.3. Any direct sum of injective abelian groups is injective.
3.4. Let p be a prime number and Z(p∞ ) the following subset of the additive group Q/Z:
Z(p∞ ) = {a/b | a, b ∈ Z, b = pi for some i ≥ 0}.
Show that
(a) Z(p∞ ) is an injective abelian group;
L
(b) Q/Z ∼
Z(p∞ );
=
p prime
(c) Z can be embedded in
Z(p∞ ) as a subgroup;
Q
p prime
(d) Q is a direct summand of
Q
Z(p∞ ).
p prime
3.5. (a) For any abelian group A and positive integer m, Hom(Zm , A) ∼
= A[m] = {a ∈ A |
ma = 0};
(b) Hom(Zm , Zn ) ∼
= Z(m,n) ;
3.6. Let R be a ring with identity, A a finitely generated R-module, and Bj , j ∈ J a family
of R-modules. Then there exists an isomorphism of abelian groups
M
M
HomR (A,
Bj ) ∼
HomR (A, Bj ).
=
j∈J
(Hint: the image of any f ∈ HomR (A,
j∈J
L
j∈J
Bj ) lies in finitely many Bj ’s.)
3.7. For any abelian group A, set Aˆ = Hom(A, Q/Z). Let m be a positive integer. Show
ˆm ∼
that Z
= Zm .
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Zero products preserving maps

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