### Abstract

For any ring A, the group K_{1}A is filtered by the Whitehead determinants of invertible matrices over A of different sizes. We want to compute the corresponding graded group (especially the highest degree non-zero term) in terms of symbols which generalize Mennicke's symbol. In particular, we generalize the Bass-Milnor-Serre result which presents SK_{1}A of a Dedekind ring A via the Mennicke symbol, to an arbitrary commutative ring A satisfying the Bass second stable range condition. As an application, SK_{1}is computed for some rings of continuous functions. Some of our theorems are partially known, but we have often weakened hypotheses, using stable range conditions rather than Krull dimension (having in mind applications to rings of continuous functions).

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

Pages (from-to) | 611-656 |

Number of pages | 46 |

Journal | Communications in Algebra |

Volume | 15 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1987 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

}

_{1}via mennicke symbols',

*Communications in Algebra*, vol. 15, no. 3, pp. 611-656. https://doi.org/10.1080/00927878708823434

**Computation of k _{1}via mennicke symbols.** / Vaserstein, Leonid N.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Computation of k1via mennicke symbols

AU - Vaserstein, Leonid N.

PY - 1987/1

Y1 - 1987/1

N2 - For any ring A, the group K1A is filtered by the Whitehead determinants of invertible matrices over A of different sizes. We want to compute the corresponding graded group (especially the highest degree non-zero term) in terms of symbols which generalize Mennicke's symbol. In particular, we generalize the Bass-Milnor-Serre result which presents SK1A of a Dedekind ring A via the Mennicke symbol, to an arbitrary commutative ring A satisfying the Bass second stable range condition. As an application, SK1is computed for some rings of continuous functions. Some of our theorems are partially known, but we have often weakened hypotheses, using stable range conditions rather than Krull dimension (having in mind applications to rings of continuous functions).

AB - For any ring A, the group K1A is filtered by the Whitehead determinants of invertible matrices over A of different sizes. We want to compute the corresponding graded group (especially the highest degree non-zero term) in terms of symbols which generalize Mennicke's symbol. In particular, we generalize the Bass-Milnor-Serre result which presents SK1A of a Dedekind ring A via the Mennicke symbol, to an arbitrary commutative ring A satisfying the Bass second stable range condition. As an application, SK1is computed for some rings of continuous functions. Some of our theorems are partially known, but we have often weakened hypotheses, using stable range conditions rather than Krull dimension (having in mind applications to rings of continuous functions).

UR - http://www.scopus.com/inward/record.url?scp=84950070007&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84950070007&partnerID=8YFLogxK

U2 - 10.1080/00927878708823434

DO - 10.1080/00927878708823434

M3 - Article

AN - SCOPUS:84950070007

VL - 15

SP - 611

EP - 656

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 3

ER -

_{1}via mennicke symbols. Communications in Algebra. 1987 Jan;15(3):611-656. https://doi.org/10.1080/00927878708823434