Let and be rings with 1. Prove that every ideal of is of the form for some ideals and .
First let and be ideals. We know that is an additive subgroup. Now let and . Then and , so that is a two-sided ideal of .
Now let be a two-sided ideal. Define and . We certainly have . Now let . Then for some and . Then . Thus . It remains to be seen that and are ideals of and , respectively. Certainly both are nonempty. Now suppose , with for some . Now if , we have , so that . Hence is an ideal. Similarly, is an ideal.