The above is the definition of "Euclidean domain" written in Dummit and Foote, abstract algebra.
I don't understand why a "Norm" is enough, it seems to me that we require a positive norm. What will happen if $b\ne 0$ but $N(b)=0$, won't that break the definition because $N(r)$ is non-negative? In that case how will we define $N(r)$ ?
For the definition of a Euclidean domain see also Wikipedia.
There is also a definition of a norm $N$ on an integral domain $D$, namely a function $N:D\rightarrow \mathbb Z$ that satisfies the following properties: $N(a)=0$ if and only if $a=0$ and $N(a)N(b)= N(ab)$. This need not be positive. For example, $D=\Bbb Z[\sqrt{2}]$ has norm $N(a+b\sqrt{2})=a^2-2b^2$, which can be negative. So units in $R=\Bbb Z[\sqrt{2}]$ can have norm $+1$ or $-1$, see here.
On the other hand $N(b)=0$ can only happen for $b=0$, which solves your problem - see also here.
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