Paths of unimodular vectors (Q1178854)

Let \(R\) be a commutative ring with 1, \(R^ n \) the free \(R\)-module of \(n\)-columns over \(R\), \(Um(n, R)\) the set of all unimodular columns in \(R^ n\). The author introduces a graph structure on \(Um(n, R)\) connecting two columns by an edge if their dot product is 1. His Theorem 1.6 asserts that when \(n \geq \) 3 the orbit \(E(n, R)\alpha\) of any \(\alpha \in R^ n\) under the subgroup \(E(n, R) \subset GL(n, R) \) generated by elementary matrices, consists of the columns \(\beta \in R^ n\) which can be connected with \(\alpha\) by a path of even length. Then the author studies the diameter of the graph. He proves that the diameter is 2 (i.e., as small as possible) if and only if \(R\) is Boolean. For any \(R\), he connects the diameter with the word length in \(E(n, R)\).