On the unit-1-stable rank of rings of analytic functions
type: Item
instance of: scholarly article
title: On the unit-1-stable rank of rings of analytic functions
zbMATH Open document ID: 0784.46034
DOI: 10.5565/PUBLMAT_362A92_08
author: Joan Josep Carmona
author: Julià Cufí
author: Pere Menal
published in: Publicacions Matemàtiques
publication date: 1994-03-22T00:00:00Z
full work available at URL: https://eudml.org/doc/41724
review text: 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.
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 203309
zbMATH Keywords: totally reducible elements
zbMATH Keywords: unit-1-stable rank
zbMATH Keywords: unimodular pair
zbMATH Keywords: totally reducible
reviewed by: George Csordas
MaRDI profile type: Publication
OpenAlex ID: W2165882820
description: scientific article; zbMATH DE number 203309