Idéaux fermés d'une algèbre de Beurling régulière

The structure of closed ideals of a regular algebra containing the classical $A^{\infty}$ is considered. Several division and approximation results are proved and a characterization of those ideals whose intersection with $A^{\infty}$ is not $\{ 0 \}$ is obtained. A complete description of the ideals with countable hull is given, with applications to synthesis of hyperfunctions.