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 $\Hom_F(A,B)$, the set of $F$-algebra homomorphisms between them, is non-empty. We also determine when the orbit set $B^\times \backslash \Hom_\falg(A,B)$ is finite and give an explicit formula for its cardinality. A few applications of main results are given.