Commutative algebra - Department of Mathematical Sciences - old ...

74 6. FINITE MODULES
6.1.7. Corollary. Let f : M → N be a homomorphism.
(1) If M is finite, then Im f is finite.
(2) If Ker f, Im f are finite, then M is finite.
(3) If N is finite, then Cok f is finite.
(4) If Im f, Cok f are finite, then N is finite.
Proof. Use 6.1.5 on the exact sequences 3.1.8.
6.1.8. Corollary. Let M, N be modules. Then M ⊕ N is finite if and only if M
and N are finite.
Proof. Use 6.1.6.
6.1.9. Proposition. Let R be a ring and M, N finite modules. Then M ⊗R N is
finite.
Proof. Use 6.1.2 and 6.1.5. Let R m → M and R n → N be surjective. Then
R m ⊗R R n → M ⊗R N is surjective, 3.4.1.
6.1.10. Proposition. Let R be a ring and M a module. The following are equivalent.
(1) M is finite and projective.
(2) M is a direct summand in a finite free module.
Proof. Use 6.1.2 and 6.1.8.
6.1.11. Proposition. Let R be a ring and M, N finite modules.
(1) If M is projective, then HomR(M, N) is finite.
(2) If M, N are projective, then HomR(M, N) is projective.
Proof. Use 6.1.8.
6.1.12. Proposition. Let F be a finite projective module and E an injective module.
(1) F ⊗R E is injective.
(2) HomR(F, E) is injective.
Proof. (1) (2) Both modules become summands in injective modules.
6.1.13. Proposition. Let R be a ring and U a multiplicative subset. For a finite
module M the following hold:
(1) U −1 M = 0 if and only if there is a u ∈ U such that uM = 0.
(2) Ann(U −1 M) = U −1 Ann(M) in U −1 R.
Proof. Let x1, . . . , xn generate M. (1) U −1 M = 0 if and only if u1x1 = · · · =
unxn. Put u = u1 . . . un. (2) Ann(M) = Ann(x1) ∩ · · · ∩ Ann(xn). Now use (1)
and 4.3.4.
6.1.14. Proposition. Let R be a ring and Mα a family of modules. For any finite
module N there is a natural isomorphism
HomR(N,
Mα)
HomR(N, Mα)
α
Proof. A homomorphism f : N → Mα has an image in a finite sum.
6.1.15. Exercise. (1) Show that
n Z/(n) is not a finite Z-module.
α