Difference between revisions of "Skew field"

m
m (Add categories)
 
Line 3: Line 3:
 
* 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>\displaystyle a + b = b + a</math> for all <math>a, b \in S</math> (Commutativity of addition.)
+
* <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.)
Line 14: Line 14:
 
==See Also==
 
==See Also==
 
* [[Abstract algebra]]
 
* [[Abstract algebra]]
 +
 +
[[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 $S$ along with two operations, $+$ and $\cdot$ such that:

  • There are elements $1, 0 \in S$ such that $1 \cdot a = a \cdot 1 = a$ and $a + 0 = 0 + a = a$ for all $a \in S$. (Existence of additive and multiplicative identities.)
  • For each $a \in S$ other than 0, there exist elements $a^{-1}, -a \in S$ such that $a\cdot a^{-1} = a^{-1}\cdot a = 1$ and $a + (-a) = (-a) + a = 0$. (Existence of additive and multiplicative inverses.)
  • $a + b = b + a$ for all $a, b \in S$ (Commutativity of addition.)
  • $(a + b) + c = a + (b + c)$ for all $a, b, c \in S$ (Associativity of addition.)
  • $(a \cdot b )\cdot c = a \cdot (b \cdot c)$ (Associativity of multiplication.)
  • $a(b + c) = ab + ac$ and $(b + c)a = ba + ca$ (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.


See Also

Invalid username
Login to AoPS