# when a prime ideal is maximal differential ideal in a UFD

Is the prime ideal $\langle X^{2}+Y^{2}-1\rangle$ a maximal differential ideal in differential ring $\mathbb{Q}[X,Y]$ with derivatives $D(X)=Y, D(Y)= -X$?

I know there are maximal ideals like $\langle X,Y-1\rangle$, $\langle X,Y\rangle$ containing $\langle X^{2}+Y^{2}-1\rangle$ but these are not proper differential ideals. If there exists a proper differential ideal containing $\langle X^{2}+Y^{2}-1\rangle$ then it must be generated by at least two polynomials say $f$, $g$. If any one the generator is of degree 1 or 2 then it is easy to see the ideal is not proper, but how to see for higher degrees of $f$ and $g$.

https://math.stackexchange.com/questions/1751020/when-a-prime-ideal-is-maximal-differential-ideal-in-a-ufd

• Welcome to MO. It is nice that you include a link to math stackexchange. However, if you feel that you might get an answer there (as you might for this question), then it is more common to wait a few days before asking the same question here. – Sebastian Goette Apr 20 '16 at 11:17