# On the Pre-Bézout Property of Wiener Algebras on the Disc and the Half-Plane

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**New Zealand Journal of Mathematics**

**Vol.** 38, (2008), **Pages** 45-55

**R. Mortini**

Université Paul Verlaine - Metz

LMAM et Département de Mathématiques

Ile du Saulcy

F-57045 METZ

FRANCE

**A. Sasane**

Mathematics Department

London School of Economics

Houghton Street

London WC2A 2AE

UNITED KINGDOM

**Abstract** Let denote the open unit disk { *z* | |*z*| < 1 }, and denote the closed right half-plane { *s* | Re(*s*) }.

(1) Let *W*^{ + } be the Wiener algebra of the disc, that is the set of all absolutely convergent Taylor series in the open unit disk , with pointwise operations.

(2) Let *W*^{ + } be the set of all functions defined in the right half-plane that differ from the Laplace transform of a function *f*_{a} *L*^{1} by a
constant. Equipped with pointwise operations, *W*^{ + }forms a ring.

We show that the rings *W*^{ + } and *W*^{ + } are pre-Bézout rings.

**Keywords** pre-Bézout ring, Wiener algebra

**Classification** (MSC2000) Primary: 30H05; Secondary: 42A99, 93D15