Homomorphisms of a matrix ring.

Question

I have the following question, What are all possible right $R$-module $(R = \mathbb{M}_2(\mathbb{Z}))$ homomorphisms from $\mathbb{M}_2(n\mathbb{Z})$ to $\mathbb{M}_2(\mathbb{Z})$. I tried to extend maps from $n\mathbb{Z}$ to $\mathbb{Z}$ defined as $f_{n,a}(nx) = ax$ and found that these are right module homomorphisms. But are these the only ones like in case of $\mathbb{Z}$ ??

Answer 1

Since $\Bbb{Z}$ is an integral domain, the entries of the matrix $nx$ are uniquely determined by $x.$

Now, you may want to ask yourself:

Is there a homomorphism which assigns $$\begin{pmatrix} 1& 0\\ 0 &1 \end{pmatrix}\mapsto\begin{pmatrix} 0 &1 \\ 1 &0 \end{pmatrix} $$ that is defined by $f_{n,a}(nx)=ax?$