If $D$ is a euclidean domain with euclidean valuation $v$, is the set $A=\{a\in D \mid v(a)>v(1)\}\cup\{0\}$ an ideal of $D$?
I'm still thinking whether $A$ is a ring or not. Seems like I can't use the definition of rings to show that $A$ is a subring of $D$. And if it is a subring, how to know if it is an ideal?
Leave alone subrings when you have to check that some subset is an ideal. For unital rings they're different beasts.
You can neither prove the set is an ideal nor it isn't.
Two examples.
If $D=\mathbb{Z}$ with $v(a)=|a|$, then $A=\mathbb{Z}\setminus\{1,-1\}$.
If $D=k[[X]]$, with the standard valuation for formal power series, then $A$ is the set of noninvertible elements in a local ring.
In the former example the set is not and ideal; in the latter it is.