Characterising Fields Based on Ideals

Theorem

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

{0}andR.
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 IR be a non-zero ideal. Then let rI be a non-zero element of I. Since R=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.