On unimodular rows

Abstract:

We prove here, among other results, that if $({x_0}, \ldots ,{x_n})$ is a unimodular row over a commutative ring $A$, $n \geqslant 2$, $x \in A$ and \[ x \equiv {x_n}\quad \mod J(A{x_0} + \cdots + A{x_{n - 2}})\] then $({x_0}, \ldots ,{x_{n - 1}},{x_n}){ \sim _E}({x_0}, \ldots ,{x_{n - 1}},x)$.