Verify size of factor ring

Question

Let the ring $R=\left \{ \begin{bmatrix} a_{1} &a_{2} \\ a_{3}& a_{4} \end{bmatrix} \mid a_{i} \in \mathbb{Z} \right \}$

and let I be the subset of R consisting of matrices with even entries.

The size of the factor ring $R/I$ is 16.

Verify the size of the factor ring.

I am unsure how to get this question started.

Any hints would be helpful.

Thanks in advance.

Edit: I've returned to this question but remain stuck.

The solution is as given below:

Why is the size $2^{4}$? And why is the integers a,b,c,d in the set ${0,1}$? From where is this being inferred from?

Answer 1

Hint: $R/I$ is isomorphic to the ring $S$ of $2\times2$ matrices with coefficients in $\mathbb{Z}/2\mathbb{Z}$. Can you find a surjective ring homomorphism $R\to S$? Can you tell what its kernel is?