Fields and trivial ideals - Imported Knowlpedia Content

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.

Remarks

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.