Let I J be ideals in a ring R Prove that I J is also an idea

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.