Let $R$ be a Euclidean domain with Euclidean function $\delta$. Prove or disprove the following: $$A=\{a \in R \mid \delta(a) > \delta(1) \} \cup \{ 0 \}$$ is an ideal of $R$.
If I take $R=\mathbb{Z} [x]$, the ring of polynomials with integer coefficients, then this is a Euclidean ring with $\delta(f)=deg(f)$. Then for $f=x+1$ and $g=-x$, we have $\deg(f)=deg(g)=1$, so $f,g \in A$. But $f+g=x+1-x=1$, so $deg(f+g)=deg(1)=0$, so $f+g \notin A$. I think this shows that $A$ is not a subring of $R$, so it is not an ideal of $R$ either. Is my thinking correct?