Let ${𝒜}_{\beta }$ be the Beurling algebra with weight $(1+|n|{)}^{\beta }$ on the unit circle $𝕋$ and, for a closed set $E\subseteq 𝕋$, let $J_{{𝒜}_{\beta }}\left(E\right)=\{f\in {𝒜}_{\beta }:f=0\text{on}\text{a}\text{neighbourhood}\text{of}E\}$. We prove that, for $\beta \>\frac{1}{2}$, there exists a closed set $E\subseteq 𝕋$ of measure zero such that the quotient algebra ${𝒜}_{\beta }/\overline{J_{{𝒜}_{\beta }}\left(E\right)}$ is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras ${\lambda }_{\gamma }$ and the algebra $𝒜𝒞$ of absolutely continuous functions on $𝕋$, we characterize the closed sets $E\subseteq 𝕋$ for which the restriction algebras ${\lambda }_{\gamma }\left(E\right)$ and $𝒜𝒞\left(E\right)$ are generated by their idempotents.