Hi, I know that if $R$ is a ring such that every projective $R$-module finitely generated is free then $R$ has the unimodular column property.
I would like to know if there is a ring $R$ that doesn't satisfy the unimodular column property but such that every finitely generated stably free $R$-module is free.


Thanks



Unimodular column property
If "unimodular column property" means that every unimodular column can be completed to an invertible matrix, then this is equivalent to the statement that every finitely generated stably free module is free. 


For finitely generated modules "Stably free implies free" is equivalent to "$P\oplus R$ free implies $P$ free" by an easy induction. The latter is equivalent to "Every unimodular column can be completed to an invertible matrix" because "$P\oplus R$ free" is equivalent to "$P$ is the cokernel of a unimodular column", and then $P$ is free if and only if the column can be completed.


I see that darij grinberg said most of this in comments already.


