Let I and J be nonzero ideals, in a commutative ring R. If R is a domain, Prove that the intersection of I and J does not equal {0}.

I have that I and J are nonzero so I has a 0 elements and elements a,b such that a+b is also in I. The same goes for J. R is a domain so it has a 1 different from 0. Does the intersection of I and J must contain 1? If that is the case then does that mean that the intersection is nonempty?