Signatures of higher level on rings with many units (Q1122610)

In this paper, \textit{E. Becker} and \textit{A. Rosenberg}'s theory of higher level reduced forms and reduced Witt rings [J. Algebra 92, 477-503 (1985; Zbl 0555.10009)] is extended to rings with many units, in particular, semi-local rings. Specifically, we show that if A is a ring with many units and \(T\subseteq A\) is a preorder of finite even exponent k then T gives rise to a space of signatures in the sense of \textit{C. Mulcahy} [Commun. Algebra 16, No.3, 577-612 (1988; Zbl 0644.10016)]. A local- global criterion for isotropy and a representation theorem are also obtained, generalizing results of \textit{M. Knebusch} in the case \(k=2\) [Abh. Math. Semin. Univ. Hamb. 51, 149-195 (1981; Zbl 0469.10008)].