Difference between revisions of "Skew field"
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Every field is a skew field. | Every field is a skew field. | ||
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+ | The most famous example of a skew field that is not also a field is the collection of [[quaternion]]s. | ||
==See Also== | ==See Also== | ||
* [[Abstract algebra]] | * [[Abstract algebra]] |
Revision as of 12:11, 16 November 2006
A skew field, also known as a division ring, is a field in which multiplication does not necessarily commute, or alternatively a (not necessarily commutative) ring in which every element has a two-sided inverse. 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.