A linear algebra characterization of the semisimplicity and the simplicity of arbitrary algebras

TL;DR

This paper provides a linear algebra-based characterization of when an arbitrary algebra is semisimple or simple, using properties like zero annihilator and specific types of division linear bases.

Contribution

It introduces new criteria based on linear algebra for determining semisimplicity and simplicity of any algebra, regardless of dimension or base field.

Findings

An algebra is semisimple iff it has zero annihilator and a semi-division linear basis.

An algebra is simple iff it has zero annihilator and an i-division linear basis.

Provides a characterization applicable to algebras of any dimension and base field.

Abstract

We show that an arbitrary algebra ${ A}$, (of arbitrary dimension, over an arbitrary base field and any identity is not suppose for the product), is semisimple if and only if it has zero annihilator and admits a semi-division linear basis. We also show that $A$ is simple if and only if it has zero annihilator and admits an $i$-division linear basis.