Classifying complements for associative algebras

For a given extension A subset of E of associative algebras we describe and classify up to an isomorphism all A-complements of E, i.e. all subalgebras X of E such that E = A + X and A boolean AND X = {0}. Let X be a given complement and (A, X, , <-, -> the canonical matched pair associated with the factorization E = A + X. We introduce a new type of deformation of the algebra X by means of the given matched pair and prove that all A-complements of E are isomorphic to such a deformation of X. Several explicit examples involving the matrix algebra are provided. (C) 2014 Elsevier Inc. All rights reserved.