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Let $𝒜$ be a commutative complex semisimple Banach algebra. Denote by $\mathrm{kh}\left(\mathrm{soc}\right(𝒜\left)\right)$ the kernel of the hull of the socle of $𝒜$. In this work we give some new characterizations of this ideal in terms of minimal idempotents in $𝒜$. This allows us to show that a “result” from Riesz theory in commutative Banach algebras is not true.