Divisibility structure and finitely generated ideals in the disc algebra

Abstract

The main purpose of this paper is the following algebraic generalization of the corona theorem for the disc algebra : If d is a greatest common divisor of the functions , then there exist functions with d = f 1 g 1 +...+ f n g n . This generalization is false for many algebras of holomorphic functions, e. g. in case of the Banach algebra H ∞ . Under the assumption that a greatest common divisor d exists, also a description of d is given.