I need to prove that the ring of dual numbers over a field $K$, $K[\epsilon]$, defined by $a+b\epsilon$, with $a, b \in K$ has exactly three ideals. I already proved that it is isomorphic to $K[X]/(X^2)$.
I'm wondering if I need to use the correspondence theorem to prove this, or if there is another (perhaps quicker) way to do this? Thanks!