Ideals in the center of symmetric algebras

TL;DR

This paper investigates symmetric algebras over an algebraically closed field, focusing on conditions where key central ideals such as the Jacobson radical, socle, or Reynolds ideal are themselves ideals.

Contribution

It provides new insights into the structure of symmetric algebras by analyzing when certain central substructures form ideals.

Findings

Identification of conditions for the Jacobson radical of the center to be an ideal.

Characterization of when the socle of the center is an ideal.

Analysis of the Reynolds ideal as an ideal in symmetric algebras.

Abstract

We study symmetric algebras $A$ over an algebraically closed field $F$ in which the Jacobson radical of the center of $A$, the socle of the center of $A$ or the Reynolds ideal of $A$ are ideals.