Skew field

Symbol-free definition
A unital ring (not necessarily commutative) is termed a skew field if it satisfies the following equivalent conditions:


 * The multiplicative group comprises all nonzero elements; equivalently, all nonzero elements are invertible
 * The ring has no proper nontrivial left ideal
 * The ring has no proper nontrivial right ideal

Stronger properties

 * Division ring is a skew field that is finite-dimensional over its center
 * Field

Weaker properties

 * Simple ring
 * Primitive ring

Left-right symmetry
The property of being a skew field is left-right symmetric, that is, a ring is a skew field if and only if its opposite ring is a skew field.