Let R be a commutative ring and N an ideal of R. Referring to Exercise 30, show that if every element of N is nilpotent and the nilradical of R/N is R/N, then the nilradical of R is R.

Data from Exercise 30

An element a of a ring R is nilpotent if aⁿ= 0 for some n ∈ Z⁺. Show that the collection of all nilpotent elements in a commutative ring R is an ideal, the nilradical of R.