Classifying complements for associative algebras

For a given extension $A\subset E$ 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 $A\cap X=\{0\}$. Let $X$ be a given complement and $(A, , X, , \triangleright, \triangleleft, \leftharpoonup, \rightharpoonup \bigl)$ 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.