ABSTRACT
This chapter is concerned with the extending structures (ES) problem for unitary associative algebras. In this context the ES problem is precisely the categorical dual of Holder's extension problem and it also generalizes the so-called radical embedding problem introduced by Hall [127]. The abstract construction of a unified product for associative algebras is introduced together with two non-abelian cohomological type objects related to the classification part of the ES problem. On the route, a new class of algebras, called supersolvable algebras, appear as the associative algebra counterpart of supersolvable Lie algebras. All supersolvable algebras of a given dimension can be classified using a recursive type algorithm.
Several special cases of the unified product are studied in detail and the problem for which each of these products is responsible is highlighted. An important such special case is the bicrossed product of associative algebras which answers the factorization problem for algebras.
The last part of the deals with the classifying complements problem (CCP) in the context of associative algebras. The factorization index for an associative algebra extension is introduced and a formula for computing it is provided. Several explicit examples are worked out in detail.
