Abstract
Suppose R is a commutative ring with 1, б=(б ij ) is a fixedD-net of ideals of R of ordern, and Gб is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for б a homomorphismdet б of the subgroup G(б) into a certain Abelian group Φ(б). Let I be the index set {1...,n}. For each subset α⫅I let б(∝)=∑б ij б ji , wherei, ranges over all indices in α and j independently over the indices in the complement Iα (б(I) is the zero ideal). Letdet ∝(a) denote the principal minor of order |α|⩽n of the matrixa ∃ G (б) corresponding to the indices in α, and let' Φ(б) be the Cartesian product of the multiplicative groups of the quotient rings R/б(α) over all subsets α⫅ I. The homomorphismdet б is defined as follows:
It is proved that if R is a semilocal commutative Bezout ring, then the kernelKer det б coincides with the subgroup E(б) generated by all transvections in G(б). For these R is also definedTm det б.
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Literature cited
Z. I. Borevich, “Parabolic subgroups of linear groups over a semilocal ring,” Vestn. Leningr. Univ., No. 13, 16–24 (1976).
Z. I. Borevich, N. A. Vavilov, and W. Narkiewicz, “Subgroups of the full linear group over a Dedekind ring,” J. Sov. Math.,19, No. 1 (1982).
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Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 37–49, 1982.
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Borevich, Z.I., Vavilov, N.A. Determinants in net subgroups. J Math Sci 27, 2855–2865 (1984). https://doi.org/10.1007/BF01410739
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DOI: https://doi.org/10.1007/BF01410739