### Abstract

The injective stability for the general linear group modulo elementary matrices begins at one plus the stable range of the ring of entries. At one step earlier, the kernel of stabilization is perhaps larger than the group of elementary matrices. Using a Dieudonné-style determinant, it is shown that this kernel is generated by matrices of the form (1 + XY) (1 + XY), under certain conditions on the ring of entries, or in the relative case, on the ideal. For any ideal of stable rank one, the kernel is given in terms of generators (X + Z + XYZ) (X + Z + ZYX). Under somewhat stronger conditions, the kernel is shown to be a commutator subgroup.

Original language | English (US) |
---|---|

Pages (from-to) | 69-96 |

Number of pages | 28 |

Journal | Linear Algebra and Its Applications |

Volume | 95 |

Issue number | C |

DOIs | |

State | Published - Oct 1987 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

_{1}of banach algebras.

*Linear Algebra and Its Applications*,

*95*(C), 69-96. https://doi.org/10.1016/0024-3795(87)90027-9

}

_{1}of banach algebras',

*Linear Algebra and Its Applications*, vol. 95, no. C, pp. 69-96. https://doi.org/10.1016/0024-3795(87)90027-9

**Prestabilization for K _{1} of banach algebras.** / Vaserstein, L. N.; Magurnt, B. A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Prestabilization for K1 of banach algebras

AU - Vaserstein, L. N.

AU - Magurnt, B. A.

PY - 1987/10

Y1 - 1987/10

N2 - The injective stability for the general linear group modulo elementary matrices begins at one plus the stable range of the ring of entries. At one step earlier, the kernel of stabilization is perhaps larger than the group of elementary matrices. Using a Dieudonné-style determinant, it is shown that this kernel is generated by matrices of the form (1 + XY) (1 + XY), under certain conditions on the ring of entries, or in the relative case, on the ideal. For any ideal of stable rank one, the kernel is given in terms of generators (X + Z + XYZ) (X + Z + ZYX). Under somewhat stronger conditions, the kernel is shown to be a commutator subgroup.

AB - The injective stability for the general linear group modulo elementary matrices begins at one plus the stable range of the ring of entries. At one step earlier, the kernel of stabilization is perhaps larger than the group of elementary matrices. Using a Dieudonné-style determinant, it is shown that this kernel is generated by matrices of the form (1 + XY) (1 + XY), under certain conditions on the ring of entries, or in the relative case, on the ideal. For any ideal of stable rank one, the kernel is given in terms of generators (X + Z + XYZ) (X + Z + ZYX). Under somewhat stronger conditions, the kernel is shown to be a commutator subgroup.

UR - http://www.scopus.com/inward/record.url?scp=45949114964&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=45949114964&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(87)90027-9

DO - 10.1016/0024-3795(87)90027-9

M3 - Article

AN - SCOPUS:45949114964

VL - 95

SP - 69

EP - 96

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - C

ER -

_{1}of banach algebras. Linear Algebra and Its Applications. 1987 Oct;95(C):69-96. https://doi.org/10.1016/0024-3795(87)90027-9