Show that if an ideal \(I\normal R\) contains a unit then \(I = R\).
Solution
Let \(u\in I\) be a unit. Then \(u^{-1}\in R\), and the ideal property gives \[ 1=u^{-1}u\in I \] (for a left ideal; equivalently \(1=uu^{-1}\in I\) for a right ideal).
Once \(1\in I\), every \(r\in R\) satisfies \[ r=r\cdot1\in I. \] Hence \(I=R\).