Difference between revisions of "Skew field"
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| − | A '''skew field''', also known as a '''division ring''', is | + | A '''skew field''', also known as a '''division ring''', is a (not necessarily commutative) ring in which every [[element]] has a two-sided [[inverse with respect to an operation | inverse]]. Equivalently, a skew field is a [[field]] in which multiplication does not necessarily [[commutative property | commute]]. That is, it is a [[set]] <math>S</math> along with two [[operation]]s, <math>+</math> and <math>\cdot</math> such that: |
* 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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| − | Every field is a skew field. | + | Every field is a skew field. The most famous example of a skew field that is not also a field is the collection of [[quaternion]]s. |
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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.)
such that 
and 
for all 
. (Existence of additive and multiplicative 
such that 
and 
. (Existence of additive and multiplicative inverses.)
for all 
(Commutativity of addition.)
for all 
(
(Associativity of multiplication.)
and 
(The
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.
