If I and J are ideals in a ring R that is commutative with identity, and I=(a) and J=(b), prove that IJ=(ab)

So if I=(a), then I is the set of all multiples of a, and likewise J is the set of all multiples of b. So IJ={i1j1+....+ikjk} with each i a multiple of a and each j a multiple of b, and so, each product a multiple of ab. So the sum of products that are multiples of ab is a multiple of ab?

Is that correct?