Exercise 2, § 1

Let A be a ring which is not necessarily commutative and $a, p_1, \ldots, p_n$ two-sided ideals of A. Suppose that a is contained in the union of the $p_i$ and that all the $p_i$ except at most two are prime ideals (Algebra, Chapter VIII, §8, Exercise 6). Show that a is contained in one of the $p_i$.