Exercise 6, § 9

Let A be a ring.

a) Let $\mathfrak{a}$ and $\mathfrak{b}$ be two-sided ideals of A. If $\mathfrak{a}$ and $\mathfrak{b}$ are nilpotent (resp. nil ideals), then so is $\mathfrak{a}+\mathfrak{b}$.

b) Prove that the sum $\mathfrak{G}$ of all two-sided nil ideals is a two-sided nil ideal and that the sum $\mathfrak{N}$ of all nilpotent two-sided ideals is a two-sided nil ideal that contains all nilpotent ideals (VIII, p. 149, Exercise 3).

c) Prove that $\mathfrak{G}$ is the largest semiprime nil ideal (loc. cit.) of A; we say that $\mathfrak{G}$ is the nilradical of A. If A is commutative, then $\mathfrak{G}$ and $\mathfrak{N}$ are both equal to the set of nilpotent elements of A.