> [!NOTE] **Definition** (Unit Group) > > Let $(R,+,\times)$ be a [[Rings|ring with unity]]. We define the *unit group* of $R$ to be the set of [[Unit in a Ring|units]] in $R$ given by $R^{*} = \{ a \in R \mid \exists b \in R : a\times b = 1 = b\times a\} $ **Notation**: some sources use $R^{\times}$ instead of $R^{*}$. > [!Example] > The [[Unit Group of Integers Modulo n|unit group of integers modulo n]] are the set of integers coprime to $n.$ # Properties **Algebra**: Note that the unit group is indeed a [[Unit Group of Ring is a Group|group]] under multiplication.