## Some properties of ideal arithmetic

Let be a ring with 1, and let be ideals of .

- Prove that is the -smallest ideal of containing both and .
- Prove that is an ideal contained in .
- Give an example where .
- Prove that if is commutative and if , then .

- We wish to prove the following: (1) is an ideal of and (2) If is an ideal and , then .
- First we show that is an ideal of . Let for some and . Then since and are closed under subtraction. Now let . We have since and absorb on the right; similarly, . Thus is an ideal of .
- Now suppose is an ideal of with . If , then since is closed under addition, we have . Hence .

- We wish to prove that is an ideal of and that .
First, let , with and , where and . Clearly . Now let . Since is an ideal of , . Similarly, . Thus is an ideal of .

Now consider again . Since and is an ideal, , and thus . Similarly, . Thus .

- Let and let and . Evidently, , while . Thus it is not generally the case that .
- Suppose and that is commutative. We have by part (3) above.
Note that . Thus, if , we have for some and and . Then . Thus .

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## Comments

Observation: Dummit-Foote includes a note at the beginning of this section of problems that R is assumed to be a ring with identity. This fact is necessary for part 4, because otherwise (I \cap J)R =/= I \cap J

Thanks!