Involutorial division rings with arbitrary centers

Abstract

It is proved that for an arbitrary field k there exists an involutorial division ring having k as its center. The aim of this note is to prove that for any given commutative field k, there exists an involutorial division ring whose center is k. The problem whether such division rings exist has been raised by Professor Tamagawa for finite fields, and more generally for fields of finite characteristic. If the given field has characteristic 0, one can construct such division rings using differential polynomials [6]. Indeed, let K=k(x) be the field of rational functions in one indeterminate x over k, and denote its usual derivation by 6. Let R=K[t, (] be the ring of differential polynomials in t with coefficients in K written on the right, and with the product determined by at=ta+a(a, for a E K. The ring R can be identified with a ring of differential operators and the mapping J of R that sends each operator %0 t'a to its adjoint (-1)iaiti is an involution [1, p. 248]. Since R is an Ore domain, it has a field of quotients D, and the involution J can be extended (in a unique way) to an involution of D [8]. Using well-known properties of the ring of polynomials R, and the fact that k has characteristic 0, it is not hard to show that the center of D is k. It was surprisingly difficult to get the result for a field k of finite characteristic p. In this case the center of the division ring D, just constructed, is much larger than k. We also obtained centers larger than k, when we constructed other Ore domains with involution, containing k. This situation led us to the consideration of the free associative algebra k(X), on a set X of at least two elements, over k. This is an involutorial ring. Indeed, the free semigroup generated by X has an involution which is defined by mapping each word into its opposite. This map, when extended by linearity to k(X), becomes an involution of k(X). The ring Received by the editors August 27, 1971. AMS 1969 subject classifications. Primary 1658; Secondary 1646.