Exercise 11, § 8

In a ring $A$, a two-sided ideal $a$ is said to be irreducible if there exists no ordered pair of two-sided ideals $b, c$ distinct from $a$ and such that $a = b \cap c$. (a) Show that the intersection of all the irreducible two-sided ideals of $A$ reduces to 0 (note that the set of two-sided ideals containing no element $a \neq 0$ is inductive and apply Zorn’s Lemma). (b) Deduce that every two-sided ideal $a$ of $A$ is the intersection of all the irreducible ideals which contain it.