Math 254A. Homework 4 Due Wednesday, 20 September

1. Exercise 5 on page 23 (corrected): The quotient 𝒪/𝔞 of a Dedekind ring 𝒪 by an ideal 𝔞 ≠ 0 is a principal ring. Hint: For 𝔞 = 𝔭^{n} the only proper ideals of 𝒪/𝔞 are given by 𝔭/𝔭^{n}, …, 𝔭^{n}/𝔭^{n}. Choose π ∈ 𝔭 \ 𝔭^{2} and show that 𝔭^{ν} = 𝒪π^{ν} + 𝔭^{n}.

2. Exercise 8 on page 23.

3. Let A be a Dedekind ring.

(a). Let I and 𝔞 be ideals of A, with prime factorizations I = ∏_{𝔭} 𝔭^{ν(𝔭)} and 𝔞 = ∏_{𝔭} 𝔭^{α(𝔭)} . Show that I ⊇ 𝔞 if and only if ν(𝔭) ≤ α(𝔭) for all 𝔭. (b). Let 𝔞 and 𝔟 be nonzero ideals of A. Carefully prove that the ideal 𝔞 + 𝔟

equals gcd(𝔞, 𝔟) . Here the gcd is defined in the obvious way using the prime factorizations of 𝔞

and 𝔟. Indeed, write 𝔞 = ∏_{𝔭} 𝔭^{α(𝔭)} and 𝔟 = ∏_{𝔭} 𝔭^{β(𝔭)} , where the product is over all (distinct) nonzero prime ideals of A. Then gcd(𝔞, 𝔟) = ∏_{𝔭} 𝔭^{min{α(𝔭),β(𝔭)}}.

(This generalizes the fact that if a, b ∈ ℤ then gcd(a, b) is the nonnegative generator of the ideal (a, b) in ℤ.)

4. Exercise 2 on page 27. Don’t forget to prove the second sentence.