Math 515 homework #7 Due date: 19980209 D. J. Bernstein 19980202

Finite products of ideals. Fix a ring R. A finite sequence of ideals P_{1}, P_{2}, …, P_{n} has a unique product P_{1}P_{2} ⋯ P_{n}, since ideal multiplication is associative. The empty product is defined as R. Exercise 16. Let R be a Noetherian ring. Prove that there is a finite sequence of prime ideals of R, say P_{1}, P_{2}, …, P_{n}, such that P_{1}P_{2} ⋯ P_{n} = 0R. (Hint: Prove that any ideal contains a product of prime ideals.) More generally, given an ideal I, one can find prime ideals P_{1}, P_{2}, …, P_{n} such that P_{1}P_{2} ⋯ P_{n} ⊆ I ⊆ P_{1} ∩ P_{2} ∩ ⋯ ∩ P_{n}. If R = Z and I = 20Z, for example, one can take the primes 2Z, 2Z, 5Z. “Primary decomposition” is a stronger statement along the same lines.