Let $M_{2}(\mathbb{Z}_{p}[i])$ be a 2 x 2 matrix having element over the ring Gaussian Integers modulo $p$, $M_{2}(\mathbb{Z}_{p})$ be a 2 x 2 matrix over the ring of Intergers modulo $p$. And
I want to find a nontrivial surjective ring homomorphism
$$\varphi \colon M_{2}(\mathbb{Z}_{p}[i]) \longrightarrow M_{2}(\mathbb{Z}_{p})$$
between these two but I can't seem to find a definition for $\varphi$.
What could be a possible $\varphi$ for it to be a surjective ring homomorphism?
Any help would be greatly appreciated.
If $p\equiv1\pmod 4$, then $u^2+1\equiv 0\pmod p$ is soluble, so one can define $\varphi(A+iB)=A+uB$ where $A$, $B\in M_2(\Bbb Z_p)$.
If $p\equiv3\pmod 4$, then $\Bbb Z_p[i]\cong\Bbb F_{p^2}=k$, the finite field of order $p^2$. The centre of $M_2(k)$ is isomorphic to $k$, and in a surjective homomorphism $\varphi:M_2(k)\to M_2(\Bbb Z_p)$ it must be mapped into the centre of $M_2(\Bbb Z_p)$, which is isomorphic $\Bbb Z_p$. But there is no unital ring homomorphism from $k$ to $\Bbb Z_p$.
I'll leave the case $p=2$ for you to consider.