Exercise 3, § 8

We say that a (left or right) ideal $\mathfrak{a}$ of a ring is nilpotent if there exists an integer $n$ such that every product of $n$ elements of $\mathfrak{a}$ is zero.

a) Let $\mathfrak{a}$ be a nilpotent left ideal of A. Prove that the two-sided ideal $\mathfrak{a}A$ generated by $\mathfrak{a}$ is nilpotent.

b) We say that a ring A is semiprime if it does not contain any nonzero nilpotent two-sided ideals. We say that a two-sided ideal $\mathfrak{a}$ of A is semiprime if the quotient ring $A/\mathfrak{a}$ is semiprime. Prove that the intersection of a family of semiprime (two-sided) ideals is a semiprime ideal and that a semiprime ideal contains every nilpotent ideal of A.

c) A prime (two-sided) ideal (VIII, p. 133, Exercise 15) is semiprime. A two-sided ideal $\mathfrak{a}$ is semiprime if and only if it is the intersection of the prime ideals that contain it. (If $a \notin \mathfrak{a}$, consider an ideal that is maximal among the two-sided ideals containing $\mathfrak{a}$ and does not contain any powers of $a.$)