Quadratic forms, rigid elements and nonreal preorders
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- by Kazimierz Szymiczek and Joseph Yucas PDF
- Proc. Amer. Math. Soc. 88 (1983), 201-204 Request permission
Abstract:
A nonreal preorder of a quaternionic structure $q:G \times G \to B$ is a subgroup $T \subseteq G$ such that $- 1 \in T$ and $- 1 \ne t \in T$ implies $D\left \langle {1,t} \right \rangle \subseteq T$. The basic part of $q$ is defined to be the set $B = \left \{ { \pm 1} \right \} \cup \left \{ {a \in G|a\;{\text {is}}\;{\text {not}}\;2 - {\text {sided}}\;{\text {rigid}}} \right \}$. A. Carson and M. Marshall have shown that if $\left | G \right | < \infty$ then every nontrivial nonreal preorder $T$ must contain $B$. The main purpose of this note is to extend this result by replacing $\left | G \right | < \infty$ with $[G:T] < \infty$.References
- Lawrence Berman, Craig Cordes, and Roger Ware, Quadratic forms, rigid elements, and formal power series fields, J. Algebra 66 (1980), no. 1, 123–133. MR 591247, DOI 10.1016/0021-8693(80)90114-3
- Andrew B. Carson and Murray A. Marshall, Decomposition of Witt rings, Canadian J. Math. 34 (1982), no. 6, 1276–1302. MR 678670, DOI 10.4153/CJM-1982-089-1
- Murray Marshall, Abstract Witt rings, Queen’s Papers in Pure and Applied Mathematics, vol. 57, Queen’s University, Kingston, Ont., 1980. MR 674651
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 88 (1983), 201-204
- MSC: Primary 10C05
- DOI: https://doi.org/10.1090/S0002-9939-1983-0695240-0
- MathSciNet review: 695240