Subnormalizer of net subgroups in the general linear group over a ring

Abstract

Let Λ be a commutative ring in which the elements of the form ε²−1, ε∈Λ* generate the unit ideal and assume that a is any D-net of ideals of Λ of order n. It is shown that the normalizerN(б) of the net subgroup G(σ) (RZhMat, 1977, 2A280) coincides with its subnormalizer in GL(n, Λ). For noncommutative Λ the corresponding result is obtained under the assumptions: 1) in Λ the elements of the form ɛ — 1, where ɛ runs through all invertible elements of the center of Λ, generate the unit ideal, and 2) the subgroup G(σ) contains the group of block diagonal matrices with blocks of order ⩾2.