On the multiplication ideal of a nonassociative algebra (Q1076780)

The composition factors of a finite dimensional algebra, viewed as a left module for its multiplication algebra, correspond to the simple algebras in the Wedderburn-Artin decomposition of the multiplication algebra modulo its Jacobson radical. If the algebra has no nilpotent elements and the base field is algebraically closed, the correspondence is one-to-one. We conclude, for nilpotent free algebras, that the multiplication algebra and multiplication ideal coincide and the centroid is a commutative subalgebra of the multiplication algebra.