Let $f(x,y),g(x,y)\in\mathbb R[x,y]$ such that $\gcd(f(x,y),g(x,y))=1$.
Let also $I\triangleleft\mathbb R[x,y]$ be the ideal generated by $f(x,y), g(x,y)$.
$I=\langle f(x,y),g(x,y)\rangle$
What can we say it's true about the ideal, what properties does it have?
My guess is since the polynomials are relatively prime then $I\equiv R[x,y]$. Is this true? If not is there anything else that can be said about the properties of the ideal?