Let be a ring with . An element is called idempotent if . Assume is an idempotent in and that for all . (That is, .) Prove that and are two-sided ideals in and that . Show that and are identities in and respectively.
Note first that .
First we’ll show that is a two-sided ideal. First , so is nonempty. Let . Then . Now if , then and . Thus is a two-sided ideal of .
Next we’ll show that is a two-sided ideal. First , so is nonempty. Let ; then . Now if , then and . Thus is a two-sided ideal in .
Now suppose . Then we have for some . Now , and right multiplying by , we see that , so that . Thus , and so is trivial.
Let ; since , . By the recognition theorem for direct products of groups, the map given by is a group isomorphism. Now let . Then . Thus as rings.
Note that for all , and . Thus is a multiplicative identity. Similarly, and , so that is a multiplicative identity in .