Centrally essential semigroup algebras

For a cancellative semigroup $S$ and a field $F$, we prove that the semigroup algebra $FS$ is centrally essential if and only if the group of fractions $G_S$ of the semigroup $S$ exists and the group algebra $FG_S$ of $G_S$ is centrally essential. The semigroup algebra of a cancellative semigroup is centrally essential if and only if it has the classical right ring of fractions, which is a centrally essential ring. There exist noncommutative, centrally essential semigroup algebras over fields of zero characteristic (this contrasts with the known fact that centrally essential group algebras over fields of zero characteristic are commutative).