Question

Give a non zero function from M3 (Z) to Z.

Answer 1

Define the function $f: M_3(\mathbb Z)\to \mathbb Z$ in the following way:

$f([a_{ij}]_{3\times 3})=a_{11}+a_{22}+a_{33}$, where $a_{ij}\in\mathbb Z.$

This function is non-zero because $f(E)=1+1+1=3\ne 0$ for the identity matrix $E\in M_3(\mathbb Z)$.