I have found a very simple and demonstrative construction of the ring extension, which came from non-commutative generalization of Hamilton-Caley's Theorem.

Let $R$ be a ring and $f(x) = x^m-\sum\limits_{j=0}^{m-1}f_jx^j\in R[x]$ be a monic polynomial. 
We identify ring $R$ with a subring $\tilde{R} = \{\mathrm{diag}(r,r,\ldots,r): r\in R\}\subset M_m(R)$. 

Then $f(x)$ has a root of the form
$$
\alpha=\left(\begin{array}{llllll}
0& e& 0&\ldots& 0& 0 \\
0& 0& e&\ldots& 0& 0 \\
. & . &  . & . & .& . \\
0& 0&0&\ldots& 0& e \\
f_0& f_1& f_2&\ldots& f_{m-2}& f_{m-1} \\
\end{array}
\right)
$$
That is $\alpha^m-\sum\limits_{j=0}^{m-1}f_j\alpha^j = 0\in M_m(R)$.
But we note that in general:
$f(\alpha^T)\neq 0$.

See [article][1] for proof  of non-commutative generalization of Hamilton-Caley's Theorem.


  [1]: https://www.researchgate.net/publication/303994859_Non-commutative_Hamilton-Caley%27s_Theorem_and_roots_of_characteristic_polynomials_of_skew_maximal_period_linear_recurrences_over_Galois_rings_V_International_Symposium_Current_Trends_in_Cryptography_CT "Non-commutative Hamilton-Caley's Theorem and roots of characteristic polynomials of skew maximal period linear recurrences over Galois rings"