Exercise 8, § 1

(a) Let a, b be two relatively prime ideals of a ring A. Show that a: b = a, b: a = b. If c is an ideal of A such that ac ⊂ b, then c ⊂ b.

(b) Let p, q be two prime ideals neither of which is contained in the other; then p: q = p and q: p = q Give an example of two principal prime ideals p, q in the polynomial ring A = K[X, Y ] (where K is a field), neither of which is contained in the other and which are not relatively prime.

(c) Let a be an element which is not a divisor of 0 in A. Show that, if the principal ideal p = Aa is prime, the relation p = bc for two ideals b, c of A implies b = A or c = A.