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