Double Vector Spaces Over Division Rings

Abstract

Introduction. Let M be a (left) vector space over the division ring D. Let E be an arbitrary ring with an idenitity element, and suppose there is given an antihomomorphism of E into the ring of D-linear transformations in M, such that the identity element of E is mapped into the identity transformation. If we denote by E* the ring of right multiplications x ->e*f{x -xe in E then the mapping e --e* is an anti-isomorphism of E onto E*, and the situation described above amounts to having a representation of E* as D-linear transformations in M. Under these circumstances, it is convenient to think of M as having simultaneously the structure of a left D-space and that of a right E-space, and to indicate the corresponding scalar operations as m > d m m, for d c D? and as m -* m e, for e ? E. Our above requirements may then all be absorbed in the statement that the * products should behave like ordinary products. We shall say then that M has the structure of a (D, E) -space. The theory of (D, E) -spaces may, of course, be subsumed under the theory of matrices with coefficients in D. A study of matrices with coefficients in a division ring which is relevant here was made by R. Brauer in 1941,1 but this overlaps only slightly with what we propose to do. In the special case where D and E both coincide with a commutative field F, the study of (F, F)-spaces amounts to the same as Jacobson's theory of self-representations.2 Jacobson has shown that the self-representations provide a generalization of Galois theory to general field extensions, not necessarily normal, or even separable. Part of our program is to consider this aspect in the non-commutative case. The Galois theory for division rings has recently been developed by N. Jacobson 3 and, independently, by II. Cartan.4 Their chief tool is a theorem-