Some Algebraic Properties of the Wiener-laplace Algebra

— We denote by W (C+) the set of all complex-valued functions defined in the closed right half plane C+ := {s ∈ C | Re(s) ≥ 0} that differ from the Laplace transform of functions from L(0,∞) by a constant. Equipped with pointwise operations, W (C+) forms a ring. It is known that W (C+) is a pre-Bézout ring. The following properties are shown for W (C+): W (C+) is not a GCD domain, that is, there exist functions F1, F2 in W (C+) that do not possess a greatest common divisor in W (C+). W (C+) is not coherent, and in fact, we give an example of two principal ideals whose intersection is not finitely generated. We will also observe that W (C+) is a Hermite ring, by showing that the maximal ideal space of W (C+), equipped with the Gelfand topology, is contractible.