Distinguishing simple algebras by means of polynomial identities

Our main goal is to extend one of classical Razmyslov's Theorem saying that any two simple finite-dimensional Omega-algebras over an algebraically closed field, satisfying the same polynomial identities, are isomorphic. We suggest a method that allows one to reduce problems about identities of algebras with additional structure to the identities of Omega-algebras. For the convenience of the reader, we start with a full detailed proof of Razmyslov's Theorem. Then we describe our method and its consequences for the identities of graded algebras, algebras with involution, and several others.