@DanielFischer So, can we prove it like that?
$$\langle x-1, y+x^2-1 \rangle =\langle x-1, y \rangle: $$
- $x-1=x-1+0 \cdot y \Rightarrow x-1 \in \langle x-1, y \rangle \\
y+x^2-1=(x+1)(x-1)+y \in \langle x-1, y \rangle$
$$\Rightarrow \langle x-1, y+x^2-1 \rangle \subseteq \langle x-1, y \rangle$$
- $x-1 \in \langle x-1, y+x^2-1 \rangle \\ y = (y+x^2-1) - (x-1)(x+1) \Rightarrow y \in \langle x-1, y+x^2-1 \rangle $
$$\Rightarrow \langle x-1, y+x^2-1 \rangle \supseteq \langle x-1, y \rangle$$