The universal skew field of fractions ofa tensor product of free rings

defined as the ring generated by X over D, with defining relations αx = xα for all x ∈ X, α ∈ K. In the special case D = K we write K〈X〉 for KK〈X〉; further when K is commutative, K〈X〉 is called the free K-algebra on X. It is known that DK〈X〉 is always a fir (= free ideal ring) and hence has a universal field of fractions (see Th. 2.4.1, p. 105f. and Cor. 7.5.11, p. 417 of [1]). This leaves open the question whether a tensor product DK〈X1〉⊗D DK〈X2〉 has a universal field of fractions. When D = K is commutative, we shall answer this question affirmatively in Theorem 3.1 below. This question is of some interest because the multiplication algebra of (1), that is, the subring of End(K〈X〉) generated by all left and right multiplications, has the form of such a tensor product. Our indirect approach is needed, for as we shall see, the tensor product is not even a Sylvester domain as soon as the sets Xi each have more than one element, or when the tensor product has more than two factors. Some limitation on D is also necessary because in general D ⊗K D need not be embeddable in a field; indeed, it may not even be an integral domain.