The next theorem was taken from the book "Lectures on finite fields and Galois Rings" of Zhe-Xian-Wan, I used the construction of a Galois Ring proposed by the author.
Let $R=GR(p^s,p^{sm})$ and $m\mid n$. Then, there is a Galois ring $R'=GR(p^s,p^{sn})$ which contains $R$ as a subring.
Consider for some $p,s,m\in\mathbb{N}$ with $p$ prime the next construction of $GR=(p^s,p^{sm})$ with the following definitions:
Let $\mathbb{Z}_{p^s}[x]$ the polynomial ring usually defined.
Take the function $\mu:\mathbb{Z}_{p^s}[x] \rightarrow \mathbb{F}_{p}[x]$ defined as the reduction modulo $p$ of the coefficents of $h(x)$ and denote the image under $\mu$ as $\overline{h}(x)$.
A polynomial $h(x)\in\mathbb{Z}_{p^s}[x]$ is called monic basic irreducible (or monic basic primitive) iff $\overline{h}(x)\in\mathbb{F}_p[x]$ is a monic irreducible (or monic primitive) polynomial (respectivly).
Let $I=\langle{\overline{h}(x)}\rangle$ the ideal generated by $\overline{h}(x)$ in $\mathbb{F}_p[x]$ defined on the usually way and, by the irreductibility of $\overline{h}(x)$, I is a maximal ideal.
I also proof that $GR(p^s,p^{sm)}\simeq \frac{\mathbb{Z}_{p^{s}}[x]}{ \langle{h(x)}\rangle}$.