A Unique Decomposition for Algebras with Multiplicative Basis

We consider a pair (A,B) where A is an algebra over a base field F, and B={ei}i∈I a basis of A satisfying the following property: for any i,j∈I we have eiej∈Fek for some k∈I. We show that A decomposes as A=s⊕d where s is a semisimple ideal of A, (a direct sum of simple ideals), and d is the direct sum of non-simple indecomposable ideals of A. Moreover, this decomposition is unique. We show that the ideals s and d are characterized by a new linear property. An interpretation of this result in terms of graph theory is also provided.