Exercise 34, § 2

In a commutative ring A, an ideal a is said to be quasi-prime if for every pair of ideals b, c of A, the relation b ∩ c ⊂ a implies b ⊂ a or c ⊂ a; every prime ideal is quasi-prime, and every quasi-prime ideal is irreducible. If the Chinese theorem is valid in A, now prove that every irreducible ideal is quasi-prime (Alg., Chapter VI, § 1, Exercise 25).