I can't seem to recall, or find anywhere online an answer to this.
If you have a ring $R$, and elements $a,b,c\in R$, what conditions do you need on $R$ to have $a\mid b+c\implies a\mid b$ and $a\mid c$? I don't think it holds in general for any ring, but I'm struggling to find out when it does hold - is being an integral domain enough?
Edit
What I'm really trying to do is show a specific ideal $\left(a(x,y),b(x,z)\right)$ in a polynomial ring $\mathbb C[x,y,z]$ is prime. I thought that the way to do it would be take $f,g\in\mathbb C[x,y,z]$ so that $fg=ra(x,y)+sb(x,z)$ (that is the product is in the ideal). I thought I'd show that $f$ (or $g$) would have to divide both $a(x,y)$ and $b(x,y)$ but given the comments this seems incorrect. What would be the way to go about this problem?