On the idempotent and nilpotent sum numbers of matrices over certain indecomposable rings and related concepts (Q2028180)

Consider the matrix ring \(M_{n}(R)\) of \(n\times n\) matrices over an associative ring \(R\) with unity; \(R\) is not necessarily commutative. The author is interested in determining when an element may be written as a sum of nilpotent or idempotent elements. A couple of results are the following: (1) If \(R\) is a division ring and \(x,y\in R\) do not commute, then \[A:=\left[ \begin{array}{cc} -xy & x \\ -yxy & yx \end{array} \right] \] is nilpotent but its trace \(yx-xy\) is not. (2) For all \(R\) and \(n>1\) each element of \(M_{n}(R)\) can be written in the form \( E_{1}F_{1}+ \cdots +E_{k}F_{k}\) for some \(k\), where the \(E_{i}\) and \(F_{i}\) are idempotents (the proof shows that we can take \(k\leq 3n-2\)).