> [!NOTE] **Definition** (Unit of a Ring) > Let $(R,+,\times)$ be a [[Rings|ring]]. An element $u$ is called a *unit* if there is some element $v$ in $R$ such that $uv = vu = 1$ (i.e it has a multiplicative inverse in $R$). > [!Example] > A congruence class $[a]_{n}$ is a [[Unit in Integers Modulo n|unit of the integers modulo n]] iff $a$ and $n$ are coprime # Properties **Algebra**: The set of units of a ring are a multiplicative group known as its [[Unit Group of Ring|unit group]].