Skew field
Definition
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
Definition with symbols
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Relation with other properties
Stronger properties
- Division ring is a skew field that is finite-dimensional over its center
- Field
Weaker properties
Metaproperties
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.