Exercise 33, § 2

Let A be a commutative ring. An ideal a of A is said to be primal if, in A/a, the set of zero divisors is an ideal p/a; the ideal p is then prime.

(a) Now prove that every primary ideal and every irreducible ideal is primal.

(b) Let a be an ideal of A such that the set of ideals b for which a : b ≠ a has a greatest element p; now prove that p is prime and that a is primal. Give an example of a primal ideal a for which the preceding condition is not fulfilled (cf. § 1, exerc. 2).

(c) In a noetherian ring A, characterize the primal ideals, and give an example of a non-primary primal ideal (cf. § 1, exerc. 11 b)).