# Bezout domains of stable range 1,5

- Written by V. Shchedryk
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Department of Algebra

Pidstryhach Institute for Applied Problems

of Mechanics and Mathematics

National Academy of Sciences of Ukraine

**Abstract:**

A ring *R * satisfies the **1.5-stable range condition** provided that,

*aR+bR+cR=R*

where *c≠0* implies that there exists *r∈R* such that

*(a+br)R+cR=R.*

The matrix Φ=*diag(*φ_{1},φ_{2},...,φ_{n}) is called a *d*-matrix if *φ*_{i}|*φ _{i+1}*,

*i=1,...,n-1*.

Consider the Zelisko group

**G**_{Φ}*=*{* H∈GL _{n}(R) *| ∃

*S∈GL*}.

_{n}(R) : HΦ=ΦSDenote by *U _{n}^{lw}(R)*,

*U*the groups of lower and upper unitriangular (triangular matrix with diagonal entries 1)

_{n}^{up}(R)*n×n*matrices over

*R*, respectively.

The main result of my talk is

**Theorem.** Let *R* be a commutative Bezout domain. Then the following are equivalent:

1) *R* has stable range 1.5;

2) *GL*_{2}(R)=**G**_{Φ}*U _{2}^{lw}(R)*

*U*for all

_{2}^{up}(R)*2×2*nonsingular

*d*-matrices Φ over

*R*;

3) *GL*_{n}(R)=**G**_{Φ}*U _{n}^{lw}(R)*

*U*for all nonsingular

_{n}^{up}(R)*d*-matrices Φ over

*R and n≥2*.

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