# 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 finitely generated 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 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.

Title stable isomorphism StableIsomorphism 2013-03-22 15:00:00 2013-03-22 15:00:00 CWoo (3771) CWoo (3771) 4 CWoo (3771) Definition msc 19A13 AlgebraicKTheory stably isomorphic stably free