Group of Units of a Ring
We define the group of units of a ring with identity \(1 \neq 0\) to be the group \(\mathcal{U}(R)\) of all units of \(R\), under the binary operation \(\cdot:R \times R \rightarrow R\).
(sidenote: For a field \(F\), we remark that \(\mathcal{U}(F)={F \backslash \{0\}}\), so we just denote the group of units of \(F\) by \(F^\ast\). Indeed, we observe that \(F\) is an Abelian group under \(+:F \times F \rightarrow F\), and \(F^\ast\) is an Abelian group under \(\cdot:F \times F \rightarrow F\). )