Decomposition of Elementary Transvection in Elementary Group

Abstract

Let σ = (σ_ij ) be an elementary net (elementary carpet) of additive subgroups of a commutative ring (in other words, a net without diagonal), n the order of σ, ω = (ω_ij ) the derived net with respect to σ, and Ω = (Ω_ij ) the net associated with the elementary group E(σ). It is assumed that ω ⊆ σ ⊆ Ω and Ω is the smallest (complemented) net containing σ. The main result consists in finding the decomposition of any elementary transvection t_ij(α) into the product of two matrices M₁ ∈ 〈t_ij(σ_ij), t_ji(σ_ji)〉 and M₂ ∈ G(τ), where \( \uptau =\left(\begin{array}{ll}{\varOmega}_{11}\hfill & {\upomega}_{12}\hfill \\ {}{\upomega}_{21}\hfill & {\varOmega}_{22}\hfill \end{array}\right) \).