Ideals in some Rings of Nevanlinna-Smirnov Type

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 $N^p$.