The Grothendieck ring of varieties over a field $k$ is the abelian group generated by isomorphim classes $[X]$ of separated, reduced $k$-schemes $X$ of finite type with the relation
$[X]=[Y] + [X\setminus Y]$
for any $Y \subset X$ a closed immersion and with the product structure given by
$[X\times Y]= [X]\cdot[Y]$.
My question is related to the condition of separateness. Why do we need the schemes to be separated? I have been searching in different papers and notes and I couldn't find anything about it. Could the Grothendieck ring be extended to non separated schemes?
Thank you very much.