Nilpotent Elements in Rings: Is 0 the Only Nilpotent Element?

[PsychonautQQ]

Homework Statement

Show that 0 is the only in R if and only if a^2 = 0 implies a = 0.

Homework Equations

none

The Attempt at a Solution

So I'm not sure if I'm doing this right.
a^2 = a*a = 0. Therefore, either a or a is zero.

The reason I'm not sure about this is because I'm thinking about matrices, where matrix A^2 can equal zero while A doesn't equal zero.

Also, did the logic that I use only work if the original question considered the ring a domain?

 

[RUber]

For the if and only if, you should have to demonstrate the proof both ways. If there is a unique zero, then $a^2=0 \implies a=0$, and if $a^2=0 \implies a=0$, then zero is unique.