Express $(x,y)$ as a principal ideal in the ring $\mathbb{R}[x,y]/(x^2 + y^2 + 1)$.

Question

We have that $\mathbb{R}[x,y]/(x^2 + y^2 + 1)$ is a principal ideal domain but not Euclidean. How about the ideal $(x,y)$ ? Is this ideal principal?

We'd have $\displaystyle (x,y) = \big(x \mathbb{R}[x,y] + y \mathbb{R}[x,y] \big) / (x^2 + y^2 + 1)$. I don't see a single generator for this.

Answer 1

Hint:

$$x,y\in (x,y)\Rightarrow x^2+y^2\in (x,y)$$