matrices - Splitting matrix of rank one

I am having trouble understanding your English. But, if I understand you correctly, the following is a counter-example:

Let $k$ be a field and let $R$ be the ring $k[a,b,c,d]/(ab-cd)$. Then $R$ is normal and $left( begin{smallmatrix} a & c \\ d & b end{smallmatrix} right)$ has rank 1. However, we can not write this matrix as $left( begin{smallmatrix} w \\ x end{smallmatrix} right) left( begin{smallmatrix} y & z end{smallmatrix} right)$ for any $w$, $x$, $y$, $z in R$.

I think your condition should almost imply that the ring is a UFD. If I have any non-unique factorization $ab=cd$, I can use it to build a counter-example like this one.

UPDATE Here are two more examples: $R=k[a,b,c]/(ac-b^2)$ and $left( begin{smallmatrix} a & b \\ b & c end{smallmatrix} right)$.

$R=mathbb{Z}[sqrt{-5}]$ and $left( begin{smallmatrix} 2 & 1+sqrt{-5} \\ 1-sqrt{-5} & 3 end{smallmatrix} right)$.

These examples rule out most attempts I could think of to find a class of rings larger than UFDs for which the result holds.