An ideal is defined like this in my text:
Let $R$ be a commutative ring. An ideal of $R$ is a subset $I \subset R$ satisfying the following properties:
1) $0 \in I$
2) $a - b \in I$ for all $a, b \in I$
3) for all $r \in R$ and $a \in I$, we have that $ra \in I$.
But doesn't 1) follow from 2) as $a - a = 0 \in I$ for $a \in I$? Or even 3) for that matter because $0a = 0$.