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.
- 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!