Bezout rings of stable ranк 1.5 and the decomposition of a complete linear group into its multiple subgroups - Shchedrik, V. P. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
A ring $R$ is called a ring of stable rank 1.5 if, for any triple $a, b, c \in R, c ot = 0$, such that $aR + bR + cR = R$, there exists$r \in R$ such that $(a + br)R + cR = R$. It is proved that a commutative Bezout domain has a stable rank 1.5 if and only ifevery 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$$\bf{G} Φ = \{ H \in GLn(R) | \exists H_1 \in GL_n(R) : HΦ = Φ H_1\},$$where $Φ = diag (φ 1, φ 2,..., φ n), φ 1| φ 2| ... | φ n, φ n ot = 0$.