A finitely generated projective module is stably free if $P \oplus R^m \cong R^n$ for some $m,n$. Show that every stably free $R$-module is free iff every unimodular row over $R$ can be completed to a non-singular matrix.
Now, what I know is that unimodular row is essentially row of some unimodular matrix, which is of determinant 1, which means the unimodular row can be completed to a non-singular matrix weather or not the given condition satisfies. So it will be great if anyone can explain it to me, am I reading the definitions of unimodular row wrong or the question is wrong. Thanks.