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"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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Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
1 Universytetska Str., 79000 Lviv, Ukraine
Tel: (+380 322) 394 172
E-mail: oromaniv at franko.lviv.ua

Scientific seminar, August, 2017