Fields are exactly commutative division rings
A ring is a field iff it is a commutative division ring.
Fields are exactly commutative division rings: A ring is a field if and only if it is a division ring whose multiplication is commutative (equivalently, is a commutative ring in which every nonzero element is invertible).
Remarks
This lemma packages the usual equivalence of “commutative + all nonzero invertible” with the field axioms used throughout algebra and algebraic geometry.