Let I, J be ideals in a ring R. Prove that I J is also an ideal of R. Solution Answer : Let I and J be two ideals of a ring R. Then I J is an additive subgroup because intersections of sub-groups are subgroups. If f I J and x R then xf I because f I and I is an ideal and so xf J for the same reason, so xf I J. Thus , I J is an ideal of R..