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
Canonical name	StableIsomorphism
Date of creation	2013-03-22 15:00:00
Last modified on	2013-03-22 15:00:00
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	4
Author	CWoo (3771)
Entry type	Definition
Classification	msc 19A13
Related topic	AlgebraicKTheory
Defines	stably isomorphic
Defines	stably free