# 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 u _{2} . . . u_{n} || such that uA = || b_{1} b_{2} . . . b_{m} ||, where (b_{1}, b_{2}, . . ., b_{m}) is the greatest common divisor of all elements of the matrix A.*

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