# 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 ${R}^{n}$ ($n\ge 1$) 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 |