Question regarding stably free module 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.

Your definition of unimodular row is totally wrong. Rather, a unimodular row is just a row whose entries generate the unit ideal in $R$. That is, $(a_1,\dots,a_n)\in R^n$ is unimodular if there exist $b_1,\dots,b_n\in R$ such that $\sum b_ia_i=1$.