On a Transversality over Local Global Rings

The purpose of this paper prove the following property; Suppose A has many units (local global ring) and |A/m| > 5 for every maximal ideal $m{\subseteq}A$. Let(E, q) ${\in}$ Q(A) and $E=E_1{\bot}{\cdots}{\bot}E_t$ be an orthogonl decomposition of E with $t{\geq}2$ and $rk(E_i){\geq}1$, for $i=1,{\cdots},t$. Let $x{\in}E$ be a primitive vector. Then there exists ${\sigma}{\in}O(q)$ such that ${\sigma}(x)$ is transversal to this decomposition.