stable isomorphism

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 finitelygenerated free $R$ -module $R^{n}$ ( $n\\geq 1$ ) such that

M\\oplus R^{n}\\cong N\\oplus R^{n}.

A left $R$ -module is said to bestably free if it is stably isomorphic to a finitelygenerated 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 isomorphicto each other. In particular, $[M]$ is the zero element in $K_{0}(R)$ iff it is stably free.