POWERS OF SKEW AND SYMMETRIC ELEMENTS IN DIVISION RINGS

Throughout this paper D will denote a division ring with involution *, and S = (xCDIx*=x) and K = (xCDIx*=-x} will always denote the sets of symmetric and skew elements, respectively, of D. An element r 6 D is said to be a norm if r = xx * for some x 6 D; the set of all norms, N, of D is clearly a subset of S. If Z is the center of D, and ifN Z then it is known that N generates D [3,1 1];ifN C Z then [D:Z] • 4. If the characteristic of D is not 2 and if N C Z then it is trivial that SCZ. We shall carry over some results which hold for general division rings to the context of division rings with involution, where, instead of imposing the conditions globally on D we impose them on S or on K. For instance, it is easy to show that if in a division ring R, arab n = bna m, for all a,b E R, and appropriate m,n • 0 depending on a and b, then R is a field. We shall show that if we merely insist that arab n = bna m for all a,b • S then D can be at most 4-dimensional over Z, and all norms in D must be in Z. In particular, if char D- • 2, we get that all the symmetric elements must be central. If amb n = bna m for all a,b C K, we again show that D is 4-dimensional ver Z and a 2 • Z for every a • K. A result of Faith [2] says that if R is a division ring and ̂ - • R is a subring of R such that x n(x) E A, n(x) • 0, for every x • R, then R is commutative. W shall prove two analogs of this result here. We shall show that if A- • D is a subring of D and s n(s) C A, n(s) • 0, for every s • S then D is at most 4-dimensional over Z, and N C Z. If k n(k) • A for every k • K we shall show that D is at most 4-dimensional over Z and k 2 • Z for every kC K. Recall that an involution on D is said to be of the first kind if c• * = c • for every c