Vol. 95, No. 2, 1981

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Linked quaternionic mappings and their associated Witt rings

Murray Angus Marshall and Joseph Lewis Yucas

Vol. 95 (1981), No. 2, 411–425

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.

Mathematical Subject Classification
Primary: 10C05, 10C05
Secondary: 10C01
Received: 9 August 1979
Revised: 14 April 1980
Published: 1 August 1981
Murray Angus Marshall
Joseph Lewis Yucas