Abstract
In [2] it was shown that generically, finite dimensional algebras over algebraically closed fields contain no nilpotents. We show that for such an algebra the multiplication algebra and the multiplication ideal are one in the same, and that the centroid is contained in the multiplication algebra. In extending the notion of separability to general nonassociative algebras over commutative rings in [3], the author made the latter assertion for algebras that are finitely generated and projective as modules over the base ring. While this assertion does not hold in general (example 2), it does hold in the circumstances considered by that author. Let U be a finite dimensional associative algebra over the algebraically closed field K. By the Wedderburn Principal Theorem [1, p. 47] U has a subalgebra S for which S _~ U/J,
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