$\displaystyle J$ is an Euclidean Domain with $\displaystyle v:J-[0]\rightarrow\mathbb{Z}\cup[0]$. Let $\displaystyle 1$ be the multiplicative identity.Let

$\displaystyle I=[a\in{J}|v(a)>v(1)]$

Is $\displaystyle I$ an ideal?

The only thing confusing me about this definition is whether the elements $\displaystyle 0$ and $\displaystyle 1$ are in $\displaystyle I$, as if they aren't then $\displaystyle I$ isn't a subring, so it's not an ideal.

$\displaystyle 0$ isn't in the domain of $\displaystyle v$, so I'm not sure if it's in $\displaystyle I$ and $\displaystyle v(1)=v(1)$, so $\displaystyle 1$ should not be in $\displaystyle I$.