The holomorphy rings of planar ternary rings with rational prime field
Let T be a planar ternary ring (PTR). The sub PTR generated by 0 and 1 is called prime PTR of T. In this paper the prime PTR of T is always assumed to be isomorphic to the field of rational numbers \({\mathbb{Q}}\), so that \({\mathbb{Q}}\) can be regarded as a sub PTR of T. Under this condition a theorem by \textit{P. Hartmann} and \textit{S. Prieß-Crampe} [Abh. Math. Semin. Univ. Hamb. 58, 149-168 (1988; Zbl 0686.51005)] states that for any ordering P of T there exists a natural, order-compatible place of T onto a subfield of \({\mathbb{R}}\) such that the push down of P equals the ordering induced by \({\mathbb{R}}\). The author states that for any P there exists exactly one such place \(\lambda\) (P). In {\S} 2 the author extends the notion of holomorphy rings to this setting. For any preordering S of T the holomorphy ring \(A_ s\) is defined as \(A_ s=\{t\in T |\lambda\) (P)(t)\(\neq \infty\) for any \(P\in X/S\}\). In Theorem 2.6 he proves that \[ A_ s=\{t\in T | \exists n\in {\mathbb{N}}\text{ such that } n+t,n-t\in S\}s=\{n/[a\setminus (s+(a^ 2+1))]| n\in {\mathbb{N}}\cup \{0\},\;a\in T^*,\;s\in S\}. \] In {\S} 3 he proves a Kadison-Dubois representation theorem for \(A_ s\) (Theorem 3.5) and applies this to show that any mapping \(\phi:A_ s\to {\mathbb{R}}\), homomorphic with respect to addition and multiplication and satisfying \(\phi (S\cap A_ s)\geq 0\) extends to a unique place \(\mu\in \{\lambda (P) |\) \(P\in X/S\}\). The final chapter 4 mainly contains a characterization of fans (4.3) and the Marshall-Brown inequality.
scientific article; zbMATH DE number 4198085
scientific article