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