Let $R$ be a ring with unity 1. Two left $R$ -modules $M$ and $N$ are said to be stably isomorphic if there exists a finitely generated free $R$ -module $R^n$ ($n\geq1$ ) such that $$M\oplus R^n\cong N\oplus R^n.$$ A left $R$ -module is said to be stably free if it is stably isomorphic to a finitely generated free $R$ -module. In other words, $M$ is stably free if $$M\oplus R^m\cong R^n$$ for some positive integers $m,n$ .

Remark In the Grothendieck group $K_0(R)$ of a ring $R$ with 1, two finitely generated projective module representatives $M$ and $N$ such that $[M]=[N]\in K_0(R)$ iff they are stably isomorphic to each other. In particular, $[M]$ is the zero element in $K_0(R)$ iff it is stably free.