Sums of commuting square-zero transformations

Sums and Products of Square-Zero Matrices

“…In [3], it is proven that an n × n matrix M over an algebraically closed field with characteristic not equal to 2 can be written as a sum of two commuting squarezero matrices if and only if M 3 = 0 and rank M ≤ n/2. For completeness, using Theorem 1, we present here a short proof of this result for matrices over division rings.…”

“…For completeness, using Theorem 1, we present here a short proof of this result for matrices over division rings. We mention that the idea of the next proof comes from [3]. is a right basis for D n .…”

“…Also, in [3] it is proven that for any two n × n commuting square-zero matrices A and B over an algebraically closed field, we have rank A + B ≤ n/2. In this article, we generalize these results to not necessarily commuting matrices over arbitrary division rings using simple techniques.…”

Sums of square-zero endomorphisms of a free module

“…The problem of decomposing an endomorphism into a sum of square-zero endomorphisms is being studied for fifty years [2,4,5,6,8,10], and there exist several well-known results on this topic. The most widely studied case is that of endomorphisms of a finite-dimensional vector space, which correspond to n × n matrices over a field.…”