Unique tensor factorization of algebras

Abstract

Tensor product decomposition of algebras is known to be non-unique in many cases. But, as will be shown here, a $\oplus$ -indecomposable, finite-dimensional $\mathbb{C}$ -algebra A has an essentially unique tensor factorization \begin{displaymath} A = A_{1} \otimes \dots \otimes A_{r} \end{displaymath} into non-trivial, $\otimes$ -indecomposable factors $A_{i}$ . Thus the semiring of isomorphism classes of finite-dimensional $\mathbb{C}$ -algebras is a polynomial semiring $\mathbb{N}[\mathcal{X}]$ . Moreover, the field $\mathbb{C}$ of complex numbers can be replaced by an arbitrary field of characteristic zero if one restricts oneself to schurian algebras.