| Abstract: | Suppose R is a commutative ring with 1,  =(
ij
) is a fixedD-net of ideals of R of ordern, and G is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for a homomorphismdet
of the subgroup G() into a certain Abelian group  (). Let I be the index set {1...,n}. For each subset   I let ( )=
ij
ji
, wherei, ranges over all indices in and j independently over the indices in the complement I ((I) is the zero ideal). Letdet
(a) denote the principal minor of order || n of the matrixa  G () corresponding to the indices in , and let' () be the Cartesian product of the multiplicative groups of the quotient rings R/() over all subsets I. The homomorphismdet
is defined as follows:
It is proved that if R is a semilocal commutative Bezout ring, then the kernelKer det
coincides with the subgroup E() generated by all transvections in G(). For these R is also definedTm det
.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 37–49, 1982. |
