How many subrings does $M_2(\mathbb{Z}/2 \mathbb{Z})$ have? I know that there are 32 possible subsets of $M_2(\mathbb{Z}/2 \mathbb{Z})$, for one to be a subring, it must be closed under multiplication and subtraction. The zero matrix and the diagonal matricies will work, so there's at least three. 

I think there are 9.
 A: The list of subrings of $M_2(\mathbb{Z}/2\mathbb{Z})$ can be found along the lines in the following paper: http://www.renyi.hu/~maroti/scorza.pdf . I did not count all subrings, but there are many more than $9$. Already the subrings with $2$ elements are given by $R=\{0,A \}$, where $A$ is a matrix with $A^2=A$ or $A^2=0$, i.e., 
$$
A=\begin{pmatrix} 1 & 0 \cr 0 & 0\end{pmatrix}, \begin{pmatrix} 0 & 1 \cr 0 & 0\end{pmatrix}, \begin{pmatrix} 0 & 0 \cr 1 & 0\end{pmatrix}, \begin{pmatrix} 0 & 0 \cr 0 & 1\end{pmatrix}, \begin{pmatrix} 1 & 1 \cr 0 & 0\end{pmatrix}, \begin{pmatrix} 1 & 0 \cr 1 & 0\end{pmatrix}, \begin{pmatrix} 1 & 0 \cr 0 & 1\end{pmatrix},\begin{pmatrix} 0 & 1 \cr 0 & 1\end{pmatrix}, \begin{pmatrix} 1 & 1 \cr 1 & 1\end{pmatrix}.
$$ 
and then there a several subrings with $4$ elements (see the above paper), and also with $8$ elements, such as
$$
R=\{ \begin{pmatrix} a & c \cr 0 & b\end{pmatrix} \mid a,b,c \in M_2(\mathbb{Z}/2\mathbb{Z}) \}, 
$$
and
$$
R=\{ \begin{pmatrix} a & 0 \cr b & c\end{pmatrix} \mid a,b,c \in M_2(\mathbb{Z}/2\mathbb{Z}) \}, 
$$
and finally $R=M_2(\mathbb{Z}/2\mathbb{Z})$.
