I have a question about ideals generated by two elements. I've searched MathStackexchange and found some related posts, but I haven't been able to understand how it all works.
The question is in general:
How can I show that an ideal generated by two prime elements, is indeed prime in a UFD.
More specific I want to show that in $\Bbb C[x,y]$ the ideals generated by $(x-i, y+3)$ and $(x+i,y+3)$ are in fact prime ideals.
My initial approach was: $\Bbb C[x,y]$ is a UFD. In a UFD every irreducible element is prime. In a UFD every prime element generates a prime ideal. But I'm not sure I can use that argument when my ideal is generated by two prime elements.
I know that I can check that $I$ is a prime ideal by checking that $R/I$ is an integral domain, but I'm not sure how to do that in my example. Can anyone show me?
I would be really grateful for any help!