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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94 8. NOETHERIAN MODULES<br />
Pro<strong>of</strong>. Use the sequence 3.2.7.<br />
8.1.8. Example. A module <strong>of</strong> finite length is noetherian.<br />
8.1.9. Exercise. (1) Show that <br />
N Z is not noetherian.<br />
(2) Show that <br />
N Z is not noetherian.<br />
(3) Show that Q is not a noetherian Z-module.<br />
8.2. Noetherian rings<br />
8.2.1. Definition. A ring R is a noetherian ring if R is a noetherian module.<br />
8.2.2. Proposition. The following conditions are equivalent.<br />
(1) R is noetherian.<br />
(2) Any ideal is finite.<br />
(3) Any increasing sequence <strong>of</strong> ideals is stationary.<br />
(4) Any nonempty subset <strong>of</strong> ideals <strong>of</strong> contains an ideal maximal for inclusion.<br />
Pro<strong>of</strong>. This is follows from 8.1.1.<br />
8.2.3. Proposition. The nilradical in a noetherian ring is nilpotent, i.e. √ 0 n = 0<br />
for some n. Some power <strong>of</strong> the radical <strong>of</strong> an ideal is contained in the ideal, i.e.<br />
√ I n ⊂ I for some n<br />
Pro<strong>of</strong>. √ 0 = (b1, . . . , bn) such that b k i = 0. Then ( aibi) nk = 0.<br />
8.2.4. Proposition. Given ideals I, J ⊂ R. Then the following are equivalent.<br />
(1) R/I, R/J are noetherian<br />
(2) R/I ∩ J is noetherian<br />
Pro<strong>of</strong>. This is a special case <strong>of</strong> 8.1.7.<br />
8.2.5. Lemma. (1) Any factor ring <strong>of</strong> a noetherian ring is a noetherian ring.<br />
(2) A principal ideal domain is noetherian.<br />
(3) If M is a noetherian R-module, then R/ Ann(M) is a noetherian ring.<br />
Pro<strong>of</strong>. (1), (2) are clear. (3) Analog <strong>of</strong> the pro<strong>of</strong> <strong>of</strong> 7.3.16. Let x ∈ M, then<br />
R/ Ann(x) Rx is noetherian. If x1, . . . , xn generate M, then Ann(M) =<br />
Ann(x1) ∩ · · · ∩ Ann(Xn). By 8.2.4 R/ Ann(M) is artinian.<br />
8.2.6. Proposition. Let R be a noetherian ring. Any finite R-module is noetherian.<br />
Pro<strong>of</strong>. Let M be a finite R-module. There is a surjective homomorphism R n →<br />
M → 0, 6.1.2. Conclusion by 8.2.3.<br />
8.2.7. Corollary. Let R be a noetherian ring. Any finite R-module is finite presented.<br />
8.2.8. Lemma. Let R be a noetherian ring and M, N noetherian modules.<br />
(1) M ⊗R N is noetherian.<br />
(2) HomR(M, N) is noetherian.<br />
Pro<strong>of</strong>. Conclusion by 6.1.8, 6.1.9, 8.2.6.