Show $\psi$ is a ring homorphism Let $K = \left\{ \begin{pmatrix} a & b \\ b & a \end{pmatrix}\bigg| \ a,b \in \mathbb{Z}\right\}\subseteq M_{2,2}(\mathbb{Z})$.
Let $\psi : K \to \mathbb{Z}$ be defined by $\psi$
 $\left(\begin{matrix} 
a & b \\ 
b & a
\end{matrix}\right) = a-b.$
I am having trouble showing $\psi$ is a ring homomorphism and what Ker $\psi$ is. 
 A: That it is a ring homomorphism is mechanical and shouldn't be too bad. 
Matrices in the kernel are given by 
$$
\begin{bmatrix}a&a\\a&a \end{bmatrix}
$$
$a\in \mathbb{Z}$
A: It is interesting to see how other rings homomorphism can be obtained, with a very similar structure.
It suffices to remark that the matrices considered here have the form:
$$aI+bJ$$
(I is the $2 \times 2$ unit matrix) where $J=\binom{0 \ 1}{1 \ 0}$ is an idempotent matrix (i.e., with $J^2=I$).
Any other idempotent matrix $K$ (see examples below), i.e., with property $K^2=I$) can replace $J$, giving a similar ring homomorphism, due to the main fact (I bypass the very easy additivity property) that 
$$(aI+bK)(a'I+b'K)=(aa'+bb')I+(ab'+ba')K \ \ \ \ \ \mapsto \ \ \ \ \ aa'+bb'-(ab'+ba'),$$
this image still being equal to the product of images, i.e.,  $$(a-b)(a'-b').$$
Examples of idempotent matrices: $K=\binom{1 \ 1}{0 \ 0}$,  $ \ K=\binom{-1 \ 0}{-1 \ 1}$, , $ \ K=\binom{\ \ 4 \ \ \ 5}{-3 \ -4}$ , $ \ K=\binom{-1 \ k}{\ \ 0 \ 1}$ for any $k$, etc. 
