Helmer’s theorem for 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 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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