Centrally Essential Semigroup Algebras

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 F G 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).

Centrally Essential Rings and Semirings

“…In this case, g ∈ Z(Z 2 (G)), whence there exists an element a ∈ Z 2 (G) such that z = (g, a) = 1. However z ∈ Z 1 (G), since a ∈ Z 2 (G), we obtain (6).…”

“…Then gz = a −1 ga ∈ g G , whence gz k = a −k ga k ∈ g G for any k ≥ 1. Therefore, the subgroup H, generated by z, satisfies (6). Now we consider two cases.…”

“…The work is based on papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The author is very grateful for the great help in editing the manuscript to Adel Abyzov, Oleg Lyubimtsev and Danil Tapkin.…”