On the Pre-bézout Property of Wiener Algebras on the Disc and the Half-plane

Let D denote the open unit disk {z ∈ C | |z| < 1}, and C+ denote the right half-plane {s ∈ C | Re(s) ≥ 0}. (1) Let W (D) be the Wiener algebra of the disc, that is the set of all absolutely convergent Taylor series in the open unit disk D, with pointwise operations. (2) Let W (C+) be the set of all functions defined in the right halfplane C+ that differ from the Laplace transform of a function fa ∈ L (0,∞) by a constant. Equipped with pointwise operations, W (C+) forms a ring. We show that the rings W (D) and W (C+) are pre-Bézout rings.