  In this paper, we present an isomorphism between the ring of general
polynomials over a division ring of degree $p$ over its center $F$ and the
group ring of the free monoid with $p^2$ variables. Using this isomorphism, we
define the characteristic polynomial of a matrix over any division algebra,
i.e. a general polynomial with one variable over the algebra whose roots are
precisely the left eigenvalues. Plus, we show how the left eigenvalues of a $4
\times 4$ matrices over any division algebra can be found by solving a general
polynomial equation of degree 6 over that algebra.
