The center of a ring is . Prove that is a subring of and that if has a 1, then . Prove also that the center of a division ring is a field.
Note first that since for all ; in particular, is nonempty. Next, if and , then . By the subgroup criterion, . Moreover, , so that ; by definition, is a subring.
If has a 1, then by definition, for all . Thus .
Now let be a division ring, and consider . If , note that by the cancellation law, the inverse of is unique; denote it by . Clearly, . Since , we have . Now let . , and since is arbitrary in , . Thus is a commutative division ring- that is, a field.