# Check whether a polynomial ideal is prime in the power series ring

I would like to know whether the ideal $I = \langle y^{2}(y^{2}-x^{2}) + w^{7}, y^{2}(y^{4}-x^{4}) + z^{7}\rangle$ is prime in $\mathbb{C}[[x,y,z,w]]$, the ring of formal power series in the indeterminates $x,y,z,w$. It seems that the generators don't factor in the power series ring. But I don't know to how show that the ideal itself is prime. Thanks for all hints & answers.

• I am not sure if this a useful comment, but you might try to look for another ideal $\mathcal{J} = \mathcal{I}$ with a simpler generators. – SomeOne Mar 23 '15 at 23:32
• These questions are the best, as Macaulay2 can't help - for now only, apparently - on power series. I will give it a thought. – ujsgeyrr1f0d0d0r0h1h0j0j_juj Mar 28 '15 at 18:00
• @SomeOne Do you have any nice examples where this simplifies such a question? – 54321user Aug 8 '16 at 20:43