  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.
