Exercise 34, § 2

In a commutative ring $A$ an ideal $a$ is called quasi-prime if for every ordered pair of ideals $b, c$ of $A$, the relation $b \cap c \subset a$ implies $b \subset a$ or $c \subset a$; every prime ideal is quasi-prime and every quasi-prime ideal is irreducible. If the Chinese remainder theorem is valid in $A$, show that every irreducible ideal is quasi-prime (Algebra, Chapter VI, § 1, Exercise 25).