Fields and trivial ideals

Fields and trivial ideals: Let $R$ be a commutative ring with $1$. Then $R$ is a field if and only if the only ideals of $R$ are $(0)$ and $R$. Equivalently, every nonzero element of $R$ is a unit.

This criterion characterizes fields among commutative rings via the lattice of ideals ; the forward direction uses that nonzero elements are units , and the reverse direction shows every nonzero principal ideal must be the whole ring. It is often paired with maximal iff quotient is a field to analyze maximal ideals .