Bezout Rings of Stable Range 1.5 and the Decomposition of a Complete Linear Group into the Product of its Subgroups

A ring R is called a ring of stable range 1.5 if, for any triple a, b, c ∈ R, c ≠ 0, such that aR + bR + cR = R, there exists r ∈ R such that (a + br)R + cR = R. It is proved that a commutative Bezout domain has a stable range 1.5 if and only if every invertible matrix A can be represented in the form A = HLU, where L, U are elements of the groups of lower and upper unitriangular matrices (triangular matrices with 1 on the diagonal) and the matrix H belongs to the group

$$ {\mathrm{G}}_{\Phi}=\left\{H\in {\mathrm{G}\mathrm{L}}_n\left.(R)\right|\exists {H}_1\in {\mathrm{G}\mathrm{L}}_n(R):H\Phi =\Phi {H}_1\right\}, $$

where Φ = diag(φ ₁, φ ₂,  … , φ _n ) ,  φ ₁|φ ₂| … |φ _n , φ _n  ≠ 0.