Bezout Rings of Stable Range 1.5

Abstract

A ring $R$ has a stable range 1.5 if, for every triple of left relatively prime nonzero elements $a, b$ and $c$ in $R$, there exists $r$ such that the elements $a+br$ and $c$ are left relatively prime. Let $R$ be a commutative Bezout domain. We prove that the matrix ring $M_2 (R)$ has the stable range 1.5 if and only if the ring $R$ has the same stable range.