TRUE or FALSE: If R is a commutative ring with unity, then the set
of units in R forms a subring
As I though if R is a ring, then the set of all units of R is not a subring of R because the zero element is not a unit. How about R is a commutative with unity?
Best Answer
You are correct that zero poses a problem. Note also that the sum of two units need not be a unit.
Example:
$R=\mathbb{Z}$. Then $1$ is a unit, but $1+1=2$ is not.
However, as others have mentioned, the set of nonzero units form a group under multiplication.