Show that \(R^\times\) need not be closed under addition.
Solution
Take any nonzero ring \(R\) with identity. Both \(1\) and \(-1\) are units, while \[ 1+(-1)=0 \] is not a unit. Hence \(R^\times\) need not be closed under addition. In particular, in \(\mathbb Z\) the units are \(\{\pm1\}\) and their sum can be \(0\) or \(2\), neither of which is a unit.