Trivialization of fans in planar ternary rings with rational prime field
Continuing his previus works [see particularly the author, Abh. Math. Semin. Univ. Hamburg 60, 1-15 (1990)] the author studies the place rings associated to the preorderings of planar ternary rings (PTRs) with rational prime field. As in the classical case, for any preordering S of such a PTR T, there exists a place \(\lambda_ S\) of T fully compatible with S such that any other place of T fully compatible with S factors through \(\lambda_ S\). He determines the corresponding place ring \(A^ S=\{t\in T|\lambda_ S(t)\neq \infty \}\), and shows that \(A^ S\) is nontrivial, if S is a nontrivial fan. Thus L. Bröcker's theorem on the trivialization of fans also applies to his setting: For any fan S of PTR T with rational prime field there exists a place \(\mu: T\to T'\cup \{\infty \}\) such that the push down \(S':=\mu (S)\setminus \{0,\infty \}\) of S is trivial fan of \(T'.\) Together with an approximation theorem it allows him to establish some classical characterizations of fans and SAP-preorderings and, finally, Bröcker's stability formula for PTRs with rational prime fields. For the notion of planar ternary rings with rational prime number field related definitions, the concepts of orderings, preorderings, SAP- preorderings, fans and places PTRs and their compatibility are referred in the author's previous articles [see J. Geom. 31, No.1/2, 100-113 (1988; Zbl 0642.06010) and Geom. Dedicata 27, No.2, 137-151 (1988; Zbl 0649.12016)].
scientific article; zbMATH DE number 4165885