Skew-field

2020 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