Is there any property of the set of nilpotent matrices?

Question

Let k be a field, $M_nk$ its matrix ring. We denote the set of nilpotent matrices by $N$. What properties can we know about $N$? Is there any analogy of "nilradical" in the matrix ring? Could you share any ideas on this problem?

Answer 1

There are numerous nilradicals of rings, all of which are zero anyways in $M_n(k)$ since it is a simple ring.

The set of nilpotent elements in a noncommutative ring does not have many useful properties. They are not even closed under addition or multiplication, as evidenced by:

$\begin{bmatrix}0&1\\0&0\end{bmatrix}+\begin{bmatrix}0&0\\1&0\end{bmatrix}=\begin{bmatrix}0&1\\1&0\end{bmatrix}$

and $\begin{bmatrix}0& 1\\0&0\end{bmatrix}\begin{bmatrix}0&0\\1&0\end{bmatrix}=\begin{bmatrix}1&0\\0&0\end{bmatrix}$

The only properties I can come up with at all right now is that they are closed under scalar multiplication and exponentiation, both of which are pretty trivial.