Classifying complements for associative algebras

For a given extension A⊂EA⊂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+XE=A+X and A∩X={0}A∩X={0}. Let X   be a given complement and (A,X,▹,◃,↼,⇀)(A,X,▹,◃,↼,⇀) the canonical matched pair associated with the factorization E=A+XE=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.