Geodesic
Generators of Orthogonal Groups over Valuation Rings
Canadian journal of mathematics, Tome 33 (1981) no. 1, pp. 116-128

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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 . 
Ishibashi, Hiroyuki. Generators of Orthogonal Groups over Valuation Rings. Canadian journal of mathematics, Tome 33 (1981) no. 1, pp. 116-128. doi: 10.4153/CJM-1981-011-3
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