Graded embeddings of finite dimensional simple graded algebras

Let A, B be finite dimensional G-graded algebras over an algebraically closed field K with char(K)=0, where G is an abelian group, and let IdG(A) be the set of graded identities of A (resp. IdG(B)). We show that if A, B are G-simple then there is a graded embedding ϕ:A→B if and only if IdG(B)⊆IdG(A). We also give a weaker generalization for the case where A is G-semisimple and B is arbitrary.