A quaternionic mapping
is a symmetric bilinear mapping q : G × G → B, where G, B are Abelian
groups, G has exponent 2 and contains a distinguished element −1 such that
q(a,a) = q(a,−1)∀a ∈ G. Such a mapping is said to be linked if q(a,b) = q(c,d)
implies the existence of x ∈ G such that q(a,b) = q(a,x) and q(c,d) = q(c,x). The
Witt ring W(q) of such a mapping q can be defined to be the integral group ring
Z[G] factored by the ideal generated by 1 + (−1) and the elements (a + b) − (c + d)
such that ab = cd and q(a,b) = q(c,d). If q is the quaternionic mapping associated to
a field or semi-local ring A with 2 ∈ A∙, then q is linked, and W(q) is the
Witt ring of free bilinear spaces over A. This paper gives a ring-theoretic
description of the class of rings W(q), q linked. In particular, all such rings are
shown to be strongly representational in the terminology of Kleinstein and
Rosenberg.
