if an ideal A contains 1, then A = R, so that's not the right avenue to pursue. what you know is that there is at least one non-zero element a in I, and one non-zero element b in J (we cannot, in general, assume two such elements of I or J. for example, R might be Z4, and I might be {0,2}).

what i suggest, is to consider the element ab of R, and use the fact that R is a domain.