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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Quadratic forms, rigid elements and nonreal preorders

Authors: Kazimierz Szymiczek and Joseph Yucas
Journal: Proc. Amer. Math. Soc. 88 (1983), 201-204
MSC: Primary 10C05
MathSciNet review: 695240
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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\vert a\;{\text{is}}\;{\text{not}}\;2 - {\text{sided}}\;{\text{rigid}}} \right\}$. A. Carson and M. Marshall have shown that if $ \left\vert G \right\vert < \infty $ then every nontrivial nonreal preorder $ T$ must contain $ B$. The main purpose of this note is to extend this result by replacing $ \left\vert G \right\vert < \infty $ with $ [G:T] < \infty $.

References [Enhancements On Off] (What's this?)

  • [1] L. Berman, C. Cordes and R. Ware, Quadratic forms, rigid elements, and formal power series fields, J. Algebra 66 (1980), 123-133. MR 591247 (83f:10028)
  • [2] A. Carson and M. Marshall, Decomposition of Witt rings, preprint. MR 678670 (84b:10030)
  • [3] M. Marshall, Abstract Witt rings, Queen's Papers in Pure and Appl. Math., no. 57, Queen's Univ., Kingston, Ont., 1980. MR 674651 (84b:10032)

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Article copyright: © Copyright 1983 American Mathematical Society

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