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

• Simple ring
• Primitive ring

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.