A field is a commutative ring $(R, +, \times)$ (henceforth abbreviated simply as $R$, with multiplicative identity $1$ and additive identity $0$) which additionally has the property that every nonzero element has a multiplicative inverse: for every $r \in R$ there is $x \in R$ such that $xr = rx = 1$. Conventionally we insist that a field must have more than one element: equivalently, $0 \not = 1$.

Examples