Product ideals vs. products of ideals

EDIT 1: LaTeX doesn't love me right now, apparently

EDIT 2: Hooray for not reading super-sticky topics. *embarrassed*

For one of my problems, I need to show that if I and J are ideals of ring R, then in general $\displaystyle \{ ij : i \in I, j \in J \}$ is not an ideal.

It asks for a counterexample, but I'm hard-pressed to find one. I'm fairly confident that you can't use principal ideals, which rules out integers, gaussian integers, or integer polynomials, but I feel really uncomfortable at this point with other rings; even subrings of C seem to have really convoluted ideals. That could be my inexperience talking, though.

The rest of the question asks to show that for that example $\displaystyle \sum_v{i_vj_v}$, is an ideal; I don't think that will be very hard - it's just the first part that's got me stumped.

(A later question asks to show that IJ need not be equal to I intersect J, which I think should also fall out of the example above)