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A [[Ring|ring]] in which the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856802.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856803.png" /> are uniquely solvable. In the case of an associative ring (cf. [[Associative rings and algebras|Associative rings and algebras]]) it is sufficient to require the existence of a unit 1 and the unique solvability of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856805.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856806.png" />. A commutative associative skew-field is called a [[Field|field]]. An example of a non-commutative associative skew-field is the skew-field of quaternions, defined as the set of matrices of the form
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{{TEX|done}}
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{{MSC|16A39|12E15}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856807.png" /></td> </tr></table>
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A ''skew-field'' (or ''skew field'') is a
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[[Ring|ring]] in which the equations $ax=b$ and $ya=b$ with $a\ne 0$ are uniquely solvable. In the case of an associative ring (cf.
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[[Associative rings and algebras|Associative rings and algebras]]) it is sufficient to require the existence of a unit 1 and the unique solvability of the equations $ax=1$ and $ya=1$ for any $a\ne 0$. A commutative associative skew-field is called a
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[[Field|field]]. An example of a non-commutative associative skew-field is the skew-field of quaternions, defined as the set of matrices of the form
  
over the field of complex numbers with the usual operations (see [[Quaternion|Quaternion]]). An example of a non-associative skew-field is the [[Cayley–Dickson algebra|Cayley–Dickson algebra]], consisting of all matrices of the same form as above over the skew-field of quaternions. This skew-field is alternative (see [[Alternative rings and algebras|Alternative rings and algebras]]). Any skew-field is a [[Division algebra|division algebra]] either over the field of rational numbers or over a field of residues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856808.png" />. The skew-field of quaternions is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s0856809.png" />-dimensional algebra over the field of real numbers, while the Cayley–Dickson algebra is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s08568010.png" />-dimensional. The dimension of any algebra with division over the field of real numbers is equal to 1, 2, 4, or 8 (see [[#References|[1]]], and also [[Topological ring|Topological ring]]). The fields of real or complex numbers and the skew-field of quaternions are the only connected locally compact associative skew-fields (see [[#References|[5]]]). Any finite-dimensional algebra without zero divisors is a skew-field. Any finite associative skew-field is commutative (see [[#References|[6]]], [[#References|[8]]]). An associative skew-field is characterized by the property that any non-zero module over it is free. Any non-associative skew-field is finite-dimensional [[#References|[3]]]. A similar result applies to Mal'tsev skew-fields [[#References|[7]]] (see [[Mal'tsev algebra|Mal'tsev algebra]]) and to Jordan skew-fields [[#References|[4]]] (see [[Jordan algebra|Jordan algebra]]). In contrast to the commutative case, not every associative ring without zero divisors can be imbedded in a skew-field (see [[Imbedding of rings|Imbedding of rings]]).
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$$\begin{pmatrix}a & \bar b\\ -b & \bar a\end{pmatrix}$$
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over the field of complex numbers with the usual operations (see
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[[Quaternion|Quaternion]]). An example of a non-associative skew-field is the
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[[Cayley–Dickson algebra|Cayley–Dickson algebra]], consisting of all matrices of the same form as above over the skew-field of quaternions. This skew-field is alternative (see
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[[Alternative rings and algebras|Alternative rings and algebras]]). Any skew-field is a
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[[Division algebra|division algebra]] either over the field of rational numbers or over a field of residues $\F_p = \Z/(p)$. The skew-field of quaternions is a $4$-dimensional algebra over the field of real numbers, while the Cayley–Dickson algebra is $8$-dimensional. The dimension of any algebra with division over the field of real numbers is equal to 1, 2, 4, or 8 (see
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{{Cite|Ad}}, and also
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[[Topological ring|Topological ring]]). The fields of real or complex numbers and the skew-field of quaternions are the only connected locally compact associative skew-fields (see
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{{Cite|Po}}). Any finite-dimensional algebra without zero divisors is a skew-field. Any finite associative skew-field is commutative (see
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{{Cite|Sk}},
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{{Cite|He}}). An associative skew-field is characterized by the property that any non-zero module over it is free. Any non-associative skew-field is finite-dimensional
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{{Cite|ZhSlShSh}}. A similar result applies to Mal'tsev skew-fields
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{{Cite|Fi}} (see
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[[Mal'tsev algebra|Mal'tsev algebra]]) and to Jordan skew-fields
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{{Cite|Ze}} (see
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[[Jordan algebra|Jordan algebra]]). In contrast to the commutative case, not every associative ring without zero divisors can be imbedded in a skew-field (see
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[[Imbedding of rings|Imbedding of rings]]).
  
====References====
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Associative skew-fields are also known as division rings, in particular if they are finite dimensional over their centre. For the imbedding problem see
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.F. Adams,  "On the nonexistence of elements of Hopf invariant one"  ''Bulletin Amer. Math. Soc.'' , '''64''' :  5  (1958)  pp. 279–282</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.A. Zhevlakov,  A.M. Slin'ko,  I.P. Shestakov,  A.I. Shirshov,  "Rings that are nearly associative" , Acad. Press  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.I. Zel'manov,  "Jordan division algebras"  ''Algebra and Logic'' , '''18''' :  3  (1979)  pp. 175–190  ''Algebra i Logika'' , '''18''' :  3  (1979)  pp. 286–310</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of general algebra" , Moscow  (1983)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.T. Filippov,  "Central simple Mal'tsev algebras"  ''Algebra and Logic'' , '''15''' :  2  (1976)  pp. 147–151  ''Algebra i Logika'' , '''15''' :  2  (1976)  pp. 235–242</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  I.N. Herstein,  "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR></table>
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{{Cite|Co}}.
 
 
 
 
 
 
====Comments====
 
Associative skew-fields are also known as division rings, in particular if they are finite dimensional over their centre. For the imbedding problem see [[#References|[a1]]].
 
  
The theorem that the only associative division algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s08568011.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s08568012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s08568013.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085680/s08568014.png" />, the algebra of quaternions, is known as Frobenius' theorem.
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The theorem that the only associative division algebras over $\R$ are $\R$, $\C$ or $\mathbb{H}$, the algebra of quaternions, is known as Frobenius' theorem.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Cohn,  "Skew field constructions" , Cambridge Univ. Press  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.M. Cohn,  "Algebra" , '''3''' , Wiley  (1991)  pp. Chapt. 7</TD></TR></table>
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{|
 +
|-
 +
|valign="top"|{{Ref|Ad}}||valign="top"|  J.F. Adams,  "On the nonexistence of elements of Hopf invariant one"  ''Bulletin Amer. Math. Soc.'', '''64''' :  5  (1958)  pp. 279–282    {{MR|0097059}}    {{ZBL|0178.26106}}
 +
|-
 +
|valign="top"|{{Ref|Co}}||valign="top"| P.M. Cohn,  "Skew field constructions", Cambridge Univ. Press  (1977)   {{MR|0463237}}    {{ZBL|0355.16009}}
 +
|-
 +
|valign="top"|{{Ref|Co2}}||valign="top"| P.M. Cohn,  "Algebra", '''3''', Wiley  (1991)  pp. Chapt. 7   {{MR|1098018}}    {{ZBL|0719.00002}}
 +
|-
 +
|valign="top"|{{Ref|Fi}}||valign="top"|  V.T. Filippov,  "Central simple Maltsev algebras"  ''Algebra and Logic'', '''15''' :  2  (1976)  pp. 147–151  ''Algebra i Logika'', '''15''' :  2  (1976)  pp. 235–242       
 +
|-
 +
|valign="top"|{{Ref|He}}||valign="top"|  I.N. Herstein,  "Noncommutative rings", Math. Assoc. Amer.  (1968)    {{MR|0227205}}    {{ZBL|0177.05801}}
 +
|-
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|valign="top"|{{Ref|Po}}||valign="top"|  L.S. Pontryagin,  "Topological groups", Princeton Univ. Press  (1958)  (Translated from Russian)       
 +
|-
 +
|valign="top"|{{Ref|Sk}}||valign="top"|  L.A. Skornyakov,  "Elements of general algebra", Moscow  (1983)  (In Russian)    {{MR|0730941}}    {{ZBL|0528.00001}}
 +
|-
 +
|valign="top"|{{Ref|Ze}}||valign="top"|  E.I. Zelmanov,  "Jordan division algebras"  ''Algebra and Logic'', '''18''' :  3  (1979)  pp. 175–190  ''Algebra i Logika'', '''18''' :  3  (1979)  pp. 286–310    {{MR|0566787}}   
 +
|-
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|valign="top"|{{Ref|ZhSlShSh}}||valign="top"|  K.A. Zhevlakov,  A.M. Slinko,  I.P. Shestakov,  A.I. Shirshov,  "Rings that are nearly associative", Acad. Press  (1982)  (Translated from Russian)    {{MR|0668355}}    {{ZBL|0487.17001}}
 +
|-
 +
|}

Revision as of 20:22, 1 March 2012

2010 Mathematics Subject Classification: Primary: 16A39 Secondary: 12E15 [MSN][ZBL]

A skew-field (or skew field) is a ring in which the equations $ax=b$ and $ya=b$ with $a\ne 0$ are uniquely solvable. In the case of an associative ring (cf. Associative rings and algebras) it is sufficient to require the existence of a unit 1 and the unique solvability of the equations $ax=1$ and $ya=1$ for any $a\ne 0$. A commutative associative skew-field is called a field. An example of a non-commutative associative skew-field is the skew-field of quaternions, defined as the set of matrices of the form

$$\begin{pmatrix}a & \bar b\\ -b & \bar a\end{pmatrix}$$ over the field of complex numbers with the usual operations (see Quaternion). An example of a non-associative skew-field is the Cayley–Dickson algebra, consisting of all matrices of the same form as above over the skew-field of quaternions. This skew-field is alternative (see Alternative rings and algebras). Any skew-field is a division algebra either over the field of rational numbers or over a field of residues $\F_p = \Z/(p)$. The skew-field of quaternions is a $4$-dimensional algebra over the field of real numbers, while the Cayley–Dickson algebra is $8$-dimensional. The dimension of any algebra with division over the field of real numbers is equal to 1, 2, 4, or 8 (see [Ad], and also Topological ring). The fields of real or complex numbers and the skew-field of quaternions are the only connected locally compact associative skew-fields (see [Po]). Any finite-dimensional algebra without zero divisors is a skew-field. Any finite associative skew-field is commutative (see [Sk], [He]). An associative skew-field is characterized by the property that any non-zero module over it is free. Any non-associative skew-field is finite-dimensional [ZhSlShSh]. A similar result applies to Mal'tsev skew-fields [Fi] (see Mal'tsev algebra) and to Jordan skew-fields [Ze] (see Jordan algebra). In contrast to the commutative case, not every associative ring without zero divisors can be imbedded in a skew-field (see Imbedding of rings).

Associative skew-fields are also known as division rings, in particular if they are finite dimensional over their centre. For the imbedding problem see [Co].

The theorem that the only associative division algebras over $\R$ are $\R$, $\C$ or $\mathbb{H}$, the algebra of quaternions, is known as Frobenius' theorem.

References

[Ad] J.F. Adams, "On the nonexistence of elements of Hopf invariant one" Bulletin Amer. Math. Soc., 64 : 5 (1958) pp. 279–282 MR0097059 Zbl 0178.26106
[Co] P.M. Cohn, "Skew field constructions", Cambridge Univ. Press (1977) MR0463237 Zbl 0355.16009
[Co2] P.M. Cohn, "Algebra", 3, Wiley (1991) pp. Chapt. 7 MR1098018 Zbl 0719.00002
[Fi] V.T. Filippov, "Central simple Maltsev algebras" Algebra and Logic, 15 : 2 (1976) pp. 147–151 Algebra i Logika, 15 : 2 (1976) pp. 235–242
[He] I.N. Herstein, "Noncommutative rings", Math. Assoc. Amer. (1968) MR0227205 Zbl 0177.05801
[Po] L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian)
[Sk] L.A. Skornyakov, "Elements of general algebra", Moscow (1983) (In Russian) MR0730941 Zbl 0528.00001
[Ze] E.I. Zelmanov, "Jordan division algebras" Algebra and Logic, 18 : 3 (1979) pp. 175–190 Algebra i Logika, 18 : 3 (1979) pp. 286–310 MR0566787
[ZhSlShSh] K.A. Zhevlakov, A.M. Slinko, I.P. Shestakov, A.I. Shirshov, "Rings that are nearly associative", Acad. Press (1982) (Translated from Russian) MR0668355 Zbl 0487.17001
How to Cite This Entry:
Skew-field. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Skew-field&oldid=21398
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article