A "quite superfluous" characterisation of the Jacobson radical of an associative ring

TL;DR

This paper extends the understanding of the Jacobson radical to non-unital rings by showing it can be characterized as the largest superfluous ideal relative to all regular ideals, generalizing a well-known unital ring property.

Contribution

It introduces a new characterization of the Jacobson radical for non-unital rings using the concept of superfluous ideals relative to regular ideals.

Findings

The Jacobson radical is the largest superfluous ideal relative to all regular ideals in non-unital rings.

The characterization aligns with the classical unital case, extending its applicability.

Provides a unified view of the Jacobson radical across different ring types.

Abstract

It is well-known that the Jacobson radical of a unital ring is its largest superfluous right ideal. It is recorded here that the result carries over to non-unital rings provided the notion of "superfluous" is taken relative to all regular ideals.