Let be a ring and let be ideals.
- Prove that and .
- Prove that if , then .
- We show that ; the proof of the other equality is similar. Let . Then for some , , and . Then . Note that since , . Similarly, since , . By this previous exercise, .
- Let . Then and for some and . Since , . Thus , and . Let . Then where and . Again because , we have . Moreover, . Thus .