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Generators of Orthogonal Groups over Valuation Rings

Published online by Cambridge University Press:  20 November 2018

Hiroyuki Ishibashi
Hiroyuki Ishibashi*
Affiliation:
Josai University, Sakado, Saitama, Japan
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Let be a valuation ring with unit element, i.e., is a commutative ring such that for any a and b in , either a divides b or b divides a. We assume 2 is a unit of . V is an n-ary nonsingular quadratic module over , O(V) or On(V) is the orthogonal group on V, and S is the set of symmetries in O(V). We define l(σ) to be the minimal number of factors in the expression of a of O(V) as a product of symmetries on V. For the case where is a field, l(σ) has been determined by P. Scherk [6] and J. Dieudonné [1]. In [3] I have generalized the results of Scherk to orthogonal groups over valuation domains. In the present paper I generalize my results of [3] to orthogonal groups over valuation rings.

Since is a valuation ring, it is a local ring with the maximal ideal A which consists of all nonunits of .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 1 , 01 February 1981 , pp. 116 - 128
Copyright
Copyright © Canadian Mathematical Society 1981

References

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3. Ishibashi, H., Generators of an orthogonal group over a local valuation domain, J. Algebra 55 (1978), 302–307.Google Scholar
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5. O'Meara, O. T., Introduction to quadratic forms (Springer-Verlag, Berlin, Gôttingen, Heidelberg, 1963).Google Scholar
6. Scherk, P., On the decomposition of orthogonalities into symmetries, Proc. Amer. Math. Soc. 1 (1950), 481–491.Google Scholar