Unique Tensor Factorization of Loop-Resistant Algebras over a Field of Finite Characteristic

Tensor product decomposition of algebras is known to be non-unique in many cases. But we know that a -indecomposable, finite-dimensional -algebra A has an essentially unique tensor factorizationA = A1 Arinto non-trivial, -indecomposable factors Ai. Thus the semiring of isomorphism classes of finite-dimensional -algebras is a polynomial semiring [ ]. Moreover, the field of complex numbers can be replaced by an arbitrary (not necessarily algebraically closed) field of characteristic zero if we restrict ourselves to split algebras. Here, we show that the above result still holds in finite characteristics if we only consider loop-resistant algebras.