Embeddings of fields into simple algebras: Generalizations and applications

For two semi-simple algebras A and B over an arbitrary ground field F, we give a numerical criterion when HomF-alg(A,B), the set of F-algebra homomorphisms between them, is non-empty. We also determine when the orbit set B×\HomF-alg(A,B) is finite and give an explicit formula for its cardinality. A few applications of main results are given.