Let $R$ a ring. If $(a \in R \land a^2=0 \implies a=0)$ then $R$ don't have nilpotent elements (not null).

Question

¿Is the statement true?, I think is false, but i can't find a counterexample.

Answer 1

Suppose $R$ has a nilpotent element $a, a^n=0, n>2$ and $a^{n-1}=u\neq 0, u^2=a^{2n-2}=0$ since $2n-2>n$ contradiction.