"I liked the algebraic way of looking at things.
I’m additionally fascinated when the algebraic method is applied to infinite objects”.

Irwing Kaplansky

# Bezout domains of stable range 1,5

Volodymyr Shchedryk

Department of Algebra
Pidstryhach Institute for Applied Problems
of Mechanics and Mathematics
National Academy of Sciences of Ukraine

Abstract:

A ring 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(φ12,...,φn) is called a d-matrix if φi|φi+1, i=1,...,n-1.

Consider the Zelisko group

GΦ={ H∈GLn(R) | ∃ S∈GLn(R) : HΦ=ΦS }.

Denote by Unlw(R), Unup(R)  the groups of lower and upper unitriangular (triangular matrix with diagonal entries 1) 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)  GL2(R)=GΦU2lw(R)U2up(R)  for all  2×2 nonsingular d-matrices Φ over R;

3)  GLn(R)=GΦUnlw(R)Unup(R)  for all nonsingular d-matrices Φ over R and n≥2.

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