Notes on signatures on rings (Q1072581)

The author analyzes a general type of signature defined on a (possibly non-commutative) ring R with 1. These signatures generalize the signatures which correspond to higher level orderings of fields, as defined by \textit{E. Becker, J. Harman} and \textit{A. Rosenberg} [J. Reine Angew. Math. 330, 53-75 (1982; Zbl 0466.12007)]. If U is a multiplicatively closed subset of R with \(1\in U\) and \(U\cap - U=\emptyset\), a U-prime is defined to be a maximal additively and multiplicatively closed subset P with \(U\subset P\subset R\setminus (- U)\). (A \(\{\) \(1\}\)-prime is one of Harrison's infinite primes.) The author proves that every U-prime in R corresponds to a signature on R. The topological space of all signatures on R and the category of all signatures on rings are also discussed.