On Ideals of the Algebra of $p$-adic Bounded Analytic Functions on a Disk

Abstract

Let $K$ be an algebraically closed field, complete for a non-trivial ultrametric absolute value. We denote by $A$ the $K$- Banach algebra of bounded analytic functions in the unit disk $\{x\in K \mid \vert x\vert<1\}$. We study some properties of ideals of $A$. We show that maximal ideals of infinite codimension are not of finite type and that $A$ is not a Bezout ring.