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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

Helmer’s theorem for 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 R has the stable range 1.5 provided that aR + bR + cR = R with c 6= 0 implies that there exists r ∈ R such that (a + br)R + cR = R. It is proved that a commutative Bezout domain R has stable range 1.5 if and only if for every n×m matrix A, rang A > 1, over R there is a row u = || 1 u2 . . . un || such that uA = || b1 b2 . . . bm ||, where (b1, b2, . . ., bm) is the greatest common divisor of all elements of the matrix A.

 

 

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