ILLINOIS JOURNAL OF MATHEMATICS Volume 29, Number 4, Winter 1985 STABLE RANK IN HOLOMORPHIC FUNCTION ALGEBRAS BY GUSTAVO CORACH AND FERNANDO DANIEL SUAREZ Introduction The concept of the stable rank of a ring, introduced by H. Bass [1], has been very useful in treating some problems in algebraic K-theory. In this paper we show how this concept is related to the structure of a commutative Banach algebra A. For example, we show that A has a finite stable rank if its spectrum X(A) has finite dimension. First, we prove, in an analytical way, that the algebra of continuous functions on the disc A (z C" zl -< 1 ) which are holomorphic on its interior A, has stable rank 1. In Sections 2 and 3 we give other proofs of this fact, but it is convenient to have a classical proof. Furthermore, some ideas contained in it lead to the notion of punctual stability, to be developed and applied in Section 3. In Section 2 we mention some results, taken from [4], and we apply them to the study of the stable rank of some algebras of holomorphic functions. In Section 3 we introduce the concept of stability of a ring A at a point g A. In the case of a commutative Banach algebra A we relate this concept to the topological structure of some subsets of A n. As an application we prove that (X) and (X) (see definitions below) have stable rank one. Finally we give a list of some open problems. Most results about Banach algebras we prove here may be stated for topological algebras A whose unit groups A" are open and the inversion is a homeomorphism of A’. The proofs are analogous to those given here. We thank Horacio Porta for his valuable comments. Section I This section considers notations and preliminary results. In this paper, rings and algebras have identity. The group of units of A is denoted by A’. Given a ring A, aAn is unimodular if there exists bAn such that {b,a) 27=lbia 1. We denote by Un(A the unimodular elements of A n. We say Received March 24, 1983. (C) 1985 by the Board of Trustees of the University of Illinois Manufactured in the United States of America 627