Centrally Essential Semigroup Algebras

TL;DR

This paper characterizes when semigroup algebras are centrally essential, linking this property to the existence of a group of fractions and the central essentiality of the associated group algebra.

Contribution

It establishes a precise criterion for centrally essential semigroup algebras based on their group of fractions and the properties of the corresponding group algebra.

Findings

Semigroup algebra FS is centrally essential iff G_S exists and FG_S is centrally essential.

Cancellative semigroup algebra is centrally essential iff it has a classical right ring of fractions that is centrally essential.

Existence of non-commutative centrally essential semigroup algebras over fields of zero characteristic.

Abstract

For a cancellative semigroup S and a field F, it is proved 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 non-commutative 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).