Characterising Fields Based on Ideals

Theorem

A ring with identity \(R\) is a field if and only if its only ideals are

\[ \{0\} \quad \text{and} \quad R.\]

Proof

Let \(R\) be a field and let \(I\) be an ideal of \(R\). We will prove the above result be showing that any non-zero ideal must be the ring itself. As such, let \(I \trianglelefteq R\) be a non-zero ideal. Then let \(r \in I\) be a non-zero element of \(I\). Since \(R^{\ast} = R - \{0\}\) given \(R\) is a field, we have that \(r\) must be a unit and therefore by this result \(I\) must be the ring \(R\) itself.