# Difference between revisions of "Skew field"

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* There are elements <math>1, 0 \in S</math> such that <math>1 \cdot a = a \cdot 1 = a</math> and <math>a + 0 = 0 + a = a</math> for all <math>a \in S</math>. (Existence of additive and multiplicative [[identity | identities]].) | * There are elements <math>1, 0 \in S</math> such that <math>1 \cdot a = a \cdot 1 = a</math> and <math>a + 0 = 0 + a = a</math> for all <math>a \in S</math>. (Existence of additive and multiplicative [[identity | identities]].) | ||

* For each <math>a \in S</math> other than 0, there exist elements <math>a^{-1}, -a \in S</math> such that <math>a\cdot a^{-1} = a^{-1}\cdot a = 1</math> and <math>a + (-a) = (-a) + a = 0</math>. (Existence of additive and multiplicative inverses.) | * For each <math>a \in S</math> other than 0, there exist elements <math>a^{-1}, -a \in S</math> such that <math>a\cdot a^{-1} = a^{-1}\cdot a = 1</math> and <math>a + (-a) = (-a) + a = 0</math>. (Existence of additive and multiplicative inverses.) | ||

− | * <math> | + | * <math>a + b = b + a</math> for all <math>a, b \in S</math> (Commutativity of addition.) |

* <math>(a + b) + c = a + (b + c)</math> for all <math>a, b, c \in S</math> ([[Associativity]] of addition.) | * <math>(a + b) + c = a + (b + c)</math> for all <math>a, b, c \in S</math> ([[Associativity]] of addition.) | ||

* <math>(a \cdot b )\cdot c = a \cdot (b \cdot c)</math> (Associativity of multiplication.) | * <math>(a \cdot b )\cdot c = a \cdot (b \cdot c)</math> (Associativity of multiplication.) | ||

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==See Also== | ==See Also== | ||

* [[Abstract algebra]] | * [[Abstract algebra]] | ||

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+ | [[Category:Ring theory]] | ||

+ | [[Category:Field theory]] |

## Latest revision as of 18:05, 9 September 2008

A **skew field**, also known as a **division ring**, is a (not necessarily commutative) ring in which every element has a two-sided inverse. Equivalently, a skew field is a field in which multiplication does not necessarily commute. That is, it is a set along with two operations, and such that:

- There are elements such that and for all . (Existence of additive and multiplicative identities.)
- For each other than 0, there exist elements such that and . (Existence of additive and multiplicative inverses.)
- for all (Commutativity of addition.)
- for all (Associativity of addition.)
- (Associativity of multiplication.)
- and (The distributive property.)

Every field is a skew field. The most famous example of a skew field that is not also a field is the collection of quaternions.