Classifying generic algebras

This paper proposes a program for the inductive classification of the generic unitary associative algebras of dimension n, based on previous work in the field and on two new results pertaining to different aspects of the problem. The first is a diagonalization theorem, showing that the nonnilpo- tent sections of an idempotent-creating deformation can be chosen from within the direct sum of the local rings at the newly created idempotents. The second gives sufficient conditions for a “loopless” basis graph with a specified radical flag structure to determine a unique component of the structure- constant scheme Alg_n. The procedure for classifying generic algebras is then described and illustrated by determining the generic algebras of dimension six.