In a ring, does $a^2=0$ imply $a=0$?

Question

Let $R$ be a ring and $a$ be an element of $R$: $a^2 = 0$. Must it be true that $a = 0$? (Assuming $0$ is the additive identity of $R$)

Answer 1

In $\mathbb{Z}/4\mathbb{Z}$ we have $2^2=0$, yet $2\neq0$.