Skew-field

A ring in which the equations and with 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 and for any . 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

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 . The skew-field of quaternions is a -dimensional algebra over the field of real numbers, while the Cayley–Dickson algebra is -dimensional. The dimension of any algebra with division over the field of real numbers is equal to 1, 2, 4, or 8 (see [1], 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 [5]). Any finite-dimensional algebra without zero divisors is a skew-field. Any finite associative skew-field is commutative (see [6], [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 [3]. A similar result applies to Mal'tsev skew-fields [7] (see Mal'tsev algebra) and to Jordan skew-fields [4] (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).

References

[1]	J.F. Adams, "On the nonexistence of elements of Hopf invariant one" Bulletin Amer. Math. Soc. , 64 : 5 (1958) pp. 279–282
[2]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[3]	K.A. Zhevlakov, A.M. Slin'ko, I.P. Shestakov, A.I. Shirshov, "Rings that are nearly associative" , Acad. Press (1982) (Translated from Russian)
[4]	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
[5]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[6]	L.A. Skornyakov, "Elements of general algebra" , Moscow (1983) (In Russian)
[7]	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
[8]	I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968)

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 [a1].

The theorem that the only associative division algebras over are , or , the algebra of quaternions, is known as Frobenius' theorem.

References

[a1]	P.M. Cohn, "Skew field constructions" , Cambridge Univ. Press (1977)
[a2]	P.M. Cohn, "Algebra" , 3 , Wiley (1991) pp. Chapt. 7