Decomposition of $s+sqrt{-1}$ in $mathbb{Z}[sqrt{-1}]$

I think I have found an answer: we just assume that $a,b$ are non-units and $b$ is associate with an element of $mathbb{Z}$, that is $b=z$ or $b=sqrt{-1}cdot z$ for some $zinmathbb{Z}$. then we treat those two cases and we will get a contradicion, for example, if $b=z$, then: $$s+sqrt{-1}=ab=azRightarrow frac{s}{z}+frac{sqrt{-1}}{z}$$ So, $frac{1}{z}inmathbb{Z}$ as a coefficient of $sqrt{-1}$, but the units in $mathbb{Z}$ are $pm 1$, then $b=z=pm1$ which means that $b$ is a unit in $mathbb{Z}[sqrt{-1}]$ which is a contradiction to the assumption that $a,b$ are non-units. Is that solution valid?

If $s+sqrt{-1}=um$ with $minBbb Z$ and $u$ a unit, then $$ s^2+1=N(s+sqrt{-1})=N(u)N(m)=m^2$$ so $$ 1=(m-s)(m+s)$$ which means that both $m+s$ and $m-s$ must be (the same) units in $Bbb Z$. It follows that $s=0$.