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We define a new class of Banach algebras of holomorphic functions on the unit disc $𝔻$ which contains the algebras studied in [GMR2] and [GW]. To a function G of the class ${𝒜}^1\left(𝔻\right)$ nowhere vanishing in $𝔻$ we associate a Banach algebra ${𝒜}_G^n\left(𝔻\right)$ contained in the disc algebra $𝒜\left(𝔻\right)$. We prove that all closed ideals of ${𝒜}_G^n\left(𝔻\right)$ of at most countable hull are of the standard form.