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!
Best Answer
Consider the map $\mathbb{C}[x,y] \to \mathbb{C}$ given by $f(x,y) \mapsto f(i, -3)$, i.e. evaluating the polynomial at $x=i, y=-3$.
Show this is a surjective homomorphism. Show the kernel is exactly the ideal $I=(x-i, y+3)$. Then you can use the first isomorphism theorem to show that $\mathbb{C}[x,y]/I$ is a field, and thus $I$ is a maximal (and therefore prime) ideal.
In general, though, we cannot assume that starting with two prime ideals will give us a new prime ideal. For example, the ideals $(3)$ and $(5)$ are both prime in the integers, but their product is not.