Let be a ring, and let .
- Prove that the set is a right ideal and that the set is a left ideal in .
- Prove that if is a left ideal, then the set is a two-sided ideal of .
We begin with a definition and some lemmas.
Definition: Let be a ring, a subset, and . Then and .
Lemma: Let be a ring, a subset, and . Then
- If , then and .
- Let . Now . The other statement is similar.
- If , then . Thus . The other statement is similar.
Now to the main result.
First, we show that is a right ideal of . To that end, let . Note that , so that . Moreover, , so that . Now , so that . Finally, if , then , so that . Thus is a right ideal of .
The proof that is a left ideal of is analogous.
Now let be a left ideal, let , and let . Since , is nonempty. Now , so that , and . Now and , so that . Thus is a two-sided ideal of .