Let \(R\) be a ring. An element \(r\in R\) is a unit if there exists \(s\in R\) such that \(rs = sr = 1\). The set of units of \(R\) is denoted \(R^{\times}\).
If \(r\in R^{\times}\), the element \(s\) with \(rs=sr=1\) is unique: if also \(rs'=s'r=1\), then \(s'=s'(rs)=(s'r)s=s\). It is written \(r^{-1}\), and \((R^{\times}, \cdot)\) is a group, abelian when \(R\) is commutative.