On a property of the rings of polynomials over discretely normed rings (Q1311170)

A ring \(R\) is said to be a \(GE_ 2\)-ring if its general linear group \(GL_ 2(R)\) is generated by elementary and diagonal matrices. The reviewer showed [in Publ. Math., Inst. Hautes Etud. Sci. 30, 365-413 (1966; Zbl 0144.263)] that the ring of polynomials in at least two variables over a field or in at least one variable over a ring of integers is not a \(GE_ 2\)-ring. The author generalizes this result as follows. Let \(A\) be a discretely normed ring, i.e. a ring with a norm \(x \to | x| \in R\) such that \(| x| = 1\) for any unit and \(| x | \geq 2\) for any non-zero non-unit. Then \(A[x_ 1,\dots,x_ t]\) is a \(GE_ 2\)-ring if and only if \(A\) is a skew field and \(t \leq 1\). If \(R = A[x_ 1,\dots,x_ t]\) is not a \(GE_ 2\)-ring and \(GE_ 2(R)\) denotes the subgroup of \(GL_ 2(R)\) generated by the elementary and diagonal matrices, the author shows that \(GL_ 2(R)\) contains a free subgroup of rank two meeting \(GE_ 2(R)\) in 1. He also describes conditions for a matrix to belong to \(GE_ 2(R)\) in case \(A\) has a (left or) right division algorithm.