I am looking for general set of techniques to show an ideal is prime. Right now, I have the following ideal in $\mathbb{C}[X, Y, Z]$. $$(Y - iZ, X^2 + Y^2 - 1)$$ This ideal is generated by irreducible elements, and I suspect this may be prime. I'm not sure though, and would like some help.
To give some more context, I'm trying to break up the algebraic set $V(Y^2 + Z^2, X^2 +Y^2 - 1)$ into irreducible components. It would be useful if people could tell me the techniques they use to show an algebraic set is irreducible (one way is by showing the associated ideal is prime).
Thanks.