Math 514 homework #6 Due date: 19950918 D. J. Bernstein 19950908 A ring R is Noetherian if every nonempty collection of ideals of R has a maximal element. Exercise 16. Let R be a Noetherian ring. Let I_{0} ⊆ I_{1} ⊆ I_{2} ⊆ ⋯ be an ascending chain of ideals of R. Prove that the chain stabilizes: i.e., for some n, I_{n} = I_{n+1} = I_{n+2} = · · ·. Let P be an ideal of R. We say that P is prime if (1) P ≠ R; (2) if ab ∈ P then a ∈ P or b ∈ P. Exercise 17. Let R be a Noetherian ring, and let I be an R-ideal. Prove that there is a finite sequence of prime ideals P_{1}, P_{2}, …, P_{n} such that ∏ P_{i} ⊆ I ⊆ ⋂ P_{i}. (Don’t forget I = R.) Exercise 18. Prove that any field is Noetherian.