Exercise 33, § 2

Let A be a commutative ring. An ideal a of A is called primal if, in $A/a$, the set of divisors of 0 is an ideal $p/a$; the ideal p is then prime.

(*) For (commutative) Artinian rings this definition coincides with that of Algebra, Chapter VIII, §6, Exercise 20.

(a) Show 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 $a : b \neq a$ has a greatest element $p$; show that $p$ is prime and that $a$ is primal. \* Give an example of a primal ideal $a$ for which the above condition is not satisfied (cf. § 1, Exercise 2). \* (c) In a Noetherian ring $A$ characterize the primal ideals and give an example of a non-primary primal ideal (cf. § 1, Exercise 11 (b)).