Number of left and right zero divisors and nilpotents

Question

How to count nuber of left and right divisors , nilpotents in a ring of matrices $2\times2$ in $Z_p$, where p is a prime. Cant understand how to find it for general case

Answer 1

In a finite ring, every element is either a unit or a zero divisor. There are lots of posts about counting the units of $M_n(F)$ where $F$ is a finite field, so start there.

The nilpotent element portion is more tricky, but not hard to reason out for $M_2(F)$. Other than zero, a nilpotent transformation $F^2\to F^2$ is going to have to have rank $1$. Such a matrix will look like $\begin{bmatrix}x&\lambda x\\y&\lambda y\end{bmatrix}$ for some $\lambda\in F$.

Furthermore, the characteristic polynomial for the matrix has to be $x^2$, so you can insert the matrix above into the equation $x^2=0$ and count the possible outcomes.