Exercise 8, § 9

We keep the notation of Exercise 7. Let A be a left Noetherian ring.

a) Prove that $\mathfrak{N}$ is the largest nilpotent two-sided ideal of A and that it is equal to the prime radical $\mathfrak{P}$ of A.

b) Prove that every left or right nil ideal of A is nilpotent (apply Exercise 5, a) of VIII, p. 150 to the ring $A/\mathfrak{N})$. We therefore have $\mathfrak{N}=\mathfrak{P}=\mathfrak{G}$.