Centrally Essential Group Algebras

TL;DR

This paper investigates the properties of group algebras over fields that are centrally essential rings, focusing on finite groups and fields of nonzero characteristic, to understand their algebraic structure.

Contribution

It introduces the concept of centrally essential group algebras and characterizes when group algebras over finite groups are centrally essential in nonzero characteristic.

Findings

Identification of conditions for group algebras to be centrally essential

Characterization of modular group algebras over finite groups

Extension of the theory of centrally essential rings to group algebras

Abstract

A ring $R$ with center $C$ is said to be \textit{centrally essential} if the module $R_C$ is an essential extension of the module $C_C$. In the paper, we study groups whose group algebras over fields are centrally essential rings. We focus on the centrally essential modular group algebras of finite groups over fields of nonzero characteristic. V.T.Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A. A.A.Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.