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## 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?