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 . . . u_n || such that uA = || b₁ b₂ . . . b_m ||, where (b₁, b₂, . . ., b_m) is the greatest common divisor of all elements of the matrix A.