It's clear to me that if $I_1$ and $I_2$ are two relatively prime ideals of a ring $R$, then there is no such prime ideal containing $I_1+I_2$, since by definition of relatively prime ideals $I_1+I_2=R$ and by definition of a prime ideal P, $P\neq R$
I'm wondering whether the converse is also true, i.e if no prime ideal $P$ contains the ideal $I_1+I_2$, then would that imply that $I_1$ and $I_2$ are relatively prime.