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