Let be a ring.
- Prove that if and are ideals of , then is also an ideal of .
- Prove that if is an ideal of for each , an arbitrary set, then is also an ideal of .
- In this previous exercise, we showed that is a subring of . Thus it suffices to show that absorbs on the right and the left. To that end, let and . Now , so that . Likewise, . Thus , and is an ideal.
- Again, we showed in this previous exercise that is a subring of , os it suffices to show absorption. Let and . Now for each , so that for all . Thus . Hence is an ideal of .