Let be a ring. For a fixed element , define . Prove that is a subring of containing . Prove that .
(We need not assume that has a 1.)
Note that , so that . Similarly, , so . If has a 1, then since , .
Now if , then , so that , and , so that . Thus is a subring.
If , then for all , . Thus for all , , and we have .
Now if , then for all , . So , and we have .