TITLE: Does there exist injective ring homomorphism between given rings?
QUESTION [5 upvotes]: Does there exist injective ring homomorphisms? 

$1.$ ${\bf Z}[x] \to M_n(\bf Z)$
$2.$ $\dfrac{{\bf Z}[x]}{(f(x))} \rightarrow  M_n (\bf Z)$ where $\deg f(x)=n$.

REPLY [6 votes]: As you were told, this is not possible since the image of $X$ is a matrix which always satisfies a polynomial equation.
Let $f\in\mathbb Z[X]$ be monic with $\deg f=n$. Then we can define an injective ring homomorphism $\mathbb Z[X]/(f)\to M_n(\mathbb Z)$ as follows: send $X$ to $C_f$, the companion matrix of $f$. Since the minimal polynomial of $C_f$ is $f$, the kernel of the map is $(f)$.

If $f$ is not monic, then this can be wrong: there is no injective ring homomorphism from $\mathbb Z[X]/(2X+2)$ to $\mathbb Z$ since the first is not an integral domain.