Let be a ring with identity and let be a subring containing 1. Prove that if is a unit in then is also a unit in . Show by example that the converse is false.
Since , has a 1 and we have . If is a unit in , then there exists an element such that . Since is in , is a unit in as well.
The integers and the rationals are rings, with a subring, has a 1 (namely 1), and . Now is a unit since . However, by Example 1 in the text, the only units in are 1 and -1.