Related Algebraic Structures
| Ring and field axioms | |||||
| Ring | Commutative ring |
Skew field or Division ring |
Field | ||
|---|---|---|---|---|---|
| Abelian (additive) group structure |
Yes | Yes | Yes | Yes | |
| Multiplicative structure and distributivity |
Yes | Yes | Yes | Yes | |
| Commutativity of multiplication | No | Yes | No | Yes | |
| Multiplicative inverses | No | No | Yes | Yes | |
The axioms imposed above resemble the ones familiar from other algebraic structures. For example, the existence of the binary operation "·", together with its commutativity, associativity, (multiplicative) identity element and inverses are precisely the axioms for an abelian group. In other words, for any field, the subset of nonzero elements F \ {0}, also often denoted F×, is an abelian group (F×, ·) usually called multiplicative group of the field. Likewise (F, +) is an abelian group. The structure of a field is hence the same as specifying such two group structures (on the same set), obeying the distributivity.
Important other algebraic structures such as rings arise when requiring only part of the above axioms. For example, if the requirement of commutativity of the multiplication operation · is dropped, one gets structures usually called division rings or skew fields.
Read more about this topic: Field (mathematics)
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