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Homestable isomorphism

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# 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

$M\oplus R^{m}\cong R^{n}$ |

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.

Defines:

stably isomorphic, stably free

Related:

AlgebraicKTheory

Type of Math Object:

Definition

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Reference

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## Mathematics Subject Classification

19A13*no label found*

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