An ideal is proper exactly when it contains no unit.Suggest a slogan
Let \(R\) be a ring and \(I\subseteq R\) an ideal. Then \(I\neq R\) if and only if \(I\) contains no unit of \(R\).
If \(I=R\), then the unit \(1\) lies in \(I\). If \(u\in I\) is a unit, then \(1=u^{-1}u\in I\), so \(r=r\cdot1\in I\) for every \(r\in R\) and \(I=R\).