Ideals in some Rings of Nevanlinna-Smirnov Type

TL;DR

This paper investigates the ideal structure of the algebra of holomorphic functions $N^p$ in the unit disk, establishing its Corona Property and providing conditions for ideals generated by inner functions to be the entire algebra.

Contribution

It extends the ideal theory of Nevanlinna-Smirnov type rings by proving the Corona Property and an analogue of a key theorem for $N^p$.

Findings

$N^p$ has the Corona Property.

Provides sufficient conditions for ideals generated by inner functions to be the whole algebra.

Extends ideal structure results to $N^p$ rings.

Abstract

Let $N^p$ $(1<p<\infty)$ denote the algebra of holomorphic functions in the open unit disk, introduced by I.~I.~Privalov with the notation $A_q$ in [8]. Since $N^p$ becomes a ring of Nevanlinna--Smirnov type in the sense of Mortini [7], the results from [7] can be applied to the ideal structure of the ring $N^p$. In particular, we observe that $N^p$ has the Corona Property. Finally, we prove the $N^p$-analogue of the Theorem 6 in [7], which gives sufficient conditions for an ideal in $N^p$, generated by a finite number of inner functions, to be equal to the whole algebra $N^p$.