# 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.

• See books by say Lang (Algebra), T.Y. Lam (old lecture notes and later text book) or K-theory by Bass, to cite a few, for definition of unimodular row (explained by Eric Wofsey) and much more. Oct 31 '19 at 14:13

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 $$\sum b_ia_i=1$$.