In a ring R the following conditions are equivalent.

(a) R has no nonzero nilpotent elements (see Exercise 12).

(b) If a ϵ R and a² = 0, then a = 0.

Data from exercise 12

An element of a ring is nilpotent if aⁿ = 0 for some n. Prove that in a commutative ring a + b is nilpotent if a and b are. Show that this result may be false if R is not commutative.