Difference between revisions of "Skew-field"
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− | + | {{TEX|done}} | |
+ | {{MSC|16A39|12E15}} | ||
− | + | A ''skew-field'' (or ''skew field'') is a | |
+ | [[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. | ||
+ | [[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 | ||
+ | [[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 | + | $$\begin{pmatrix}a & \bar b\\ -b & \bar a\end{pmatrix}$$ |
+ | 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 $\F_p = \Z/(p)$. 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 | ||
+ | {{Cite|Ad}}, 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 | ||
+ | {{Cite|Po}}). Any finite-dimensional algebra without zero divisors is a skew-field. Any finite associative skew-field is commutative (see | ||
+ | {{Cite|Sk}}, | ||
+ | {{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 | ||
+ | {{Cite|ZhSlShSh}}. A similar result applies to Mal'tsev skew-fields | ||
+ | {{Cite|Fi}} (see | ||
+ | [[Mal'tsev algebra|Mal'tsev algebra]]) and to Jordan skew-fields | ||
+ | {{Cite|Ze}} (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]]). | ||
− | + | Associative skew-fields are also known as division rings, in particular if they are finite dimensional over their centre. For the imbedding problem see | |
− | + | {{Cite|Co}}. | |
− | |||
− | |||
− | |||
− | |||
− | Associative skew-fields are also known as division rings, in particular if they are finite dimensional over their centre. For the imbedding problem see | ||
− | The theorem that the only associative division algebras over | + | 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==== | ||
− | + | {| | |
+ | |- | ||
+ | |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}} | ||
+ | |- | ||
+ | |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}} | ||
+ | |- | ||
+ | |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 18:22, 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 |
Skew-field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skew-field&oldid=21398