On the unit-1-stable rank of rings of analytic functions
Let \(A\) be a commutative ring with unity. A pair of elements \((a_ 1,a_ 2)\in A^ 2\) is said to be unimodular, if there exists \((b_ 1,b_ 2)\in A^ 2\) such that \(a_ 1 b_ 1+ a_ 2 b_ 2=1\). A unimodular pair \((a_ 1,a_ 2)\) is said to be totally reducible, if there exists an element \(u\in A^{-1}\) such that \(a_ 1+ ua_ 2\in A^{-1}\). The ring \(A\) is said to have unit-1-stable rank, if each unimodular pair in \(A\) is totally reducible. Let \(H(U)\) denote the ring of analytic functions on an open set \(U\subseteq\mathbb{C}\). In the paper under review, the authors prove a general result which implies that \(H(U)\) does not have unit-1-stable rank. They also prove that a function \(f\in H(U)\) is totally reducible if and only if \(f\) is the zero function.
scientific article; zbMATH DE number 203309