Let R a normal domain, that is an integrally closed noetherian domain, like Dedekind domains, UFD, etc

Let A=(a_{ i j }) a matrix with elements in R and dimension n x m.

Suppose

- rank A=1 ↔ all 2 x 2 minors are =0.
- J:= ideal generated by a
_{ i j }verify (R:(R:J))=R ↔ J is not included in any prime ideal with height 1.

If R is an UFD, with the preview conditions, we can write A like product of a n x 1 vector column C=(c_{ i }) and a 1 x m vector file F=(f_{ j }), that is a_{ i j }=c_{ i }·f_{ j }.

I conjecture that is true in the general case, but I cannot make any progress.

Have you contraexemples with normal rings?

I´m grateful for your advices!