TY - JOUR

T1 - Determinants in net subgroups

AU - Borevich, Z. I.

AU - Vavilov, N. A.

N1 - Copyright: Copyright 2007 Elsevier B.V., All rights reserved.

PY - 1984/11

Y1 - 1984/11

N2 - Suppose R is a commutative ring with 1, b{cyrillic}=(b{cyrillic}ij) is a fixed D-net of ideals of R of order n, and Gb{cyrillic} is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for b{cyrillic} a homomorphism detb{cyrillic} of the subgroup G(b{cyrillic}) into a certain Abelian group Φ(b{cyrillic}). Let I be the index set {1...,n}. For each subset α{subset double equals}I let b{cyrillic}(∝)=∑b{cyrillic}ijb{cyrillic}ji, where i, ranges over all indices in α and j independently over the indices in the complement Iα (b{cyrillic}(I) is the zero ideal). Let det∝(a) denote the principal minor of order |α|≤n of the matrix a ∃ G (b{cyrillic}) corresponding to the indices in α, and let' Φ(b{cyrillic}) be the Cartesian product of the multiplicative groups of the quotient rings R/b{cyrillic}(α) over all subsets α{subset double equals} I. The homomorphism detb{cyrillic} is defined as follows:[Figure not available: see fulltext.] It is proved that if R is a semilocal commutative Bezout ring, then the kernel Ker detb{cyrillic} coincides with the subgroup E(b{cyrillic}) generated by all transvections in G(b{cyrillic}). For these R is also defined Tm detb{cyrillic}.

AB - Suppose R is a commutative ring with 1, b{cyrillic}=(b{cyrillic}ij) is a fixed D-net of ideals of R of order n, and Gb{cyrillic} is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for b{cyrillic} a homomorphism detb{cyrillic} of the subgroup G(b{cyrillic}) into a certain Abelian group Φ(b{cyrillic}). Let I be the index set {1...,n}. For each subset α{subset double equals}I let b{cyrillic}(∝)=∑b{cyrillic}ijb{cyrillic}ji, where i, ranges over all indices in α and j independently over the indices in the complement Iα (b{cyrillic}(I) is the zero ideal). Let det∝(a) denote the principal minor of order |α|≤n of the matrix a ∃ G (b{cyrillic}) corresponding to the indices in α, and let' Φ(b{cyrillic}) be the Cartesian product of the multiplicative groups of the quotient rings R/b{cyrillic}(α) over all subsets α{subset double equals} I. The homomorphism detb{cyrillic} is defined as follows:[Figure not available: see fulltext.] It is proved that if R is a semilocal commutative Bezout ring, then the kernel Ker detb{cyrillic} coincides with the subgroup E(b{cyrillic}) generated by all transvections in G(b{cyrillic}). For these R is also defined Tm detb{cyrillic}.

UR - http://www.scopus.com/inward/record.url?scp=34250140374&partnerID=8YFLogxK

U2 - 10.1007/BF01410739

DO - 10.1007/BF01410739

M3 - Article

AN - SCOPUS:34250140374

VL - 27

SP - 2855

EP - 2865

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 4

ER -