The ring $\mathrm{M}_{8k+4}(\mathbb{Z}_2)$ is nil-clean of index four

TL;DR

This paper investigates the algebraic structure of certain matrix rings over the field with two elements, proving they are nil-clean of index four, and explores matrix decompositions with specific idempotent and nilpotent properties.

Contribution

It establishes that the ring of matrices of size 8k+4 over _2 is nil-clean of index four and analyzes limitations on decomposing certain matrices into sums of idempotent and nilpotent matrices.

Findings

The ring _2^{(8k+4) imes (8k+4)} is nil-clean of index four.

Certain matrices cannot be expressed as sums of an idempotent and a nilpotent matrix.

The paper provides conditions under which matrix decompositions over _2 are impossible.

Abstract

We show that the direct sum of an odd number of matrices $$C=\left(\begin{array}{cccc} 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&1 \end{array}\right)$$ cannot be a sum $P+Q$ of matrices over $\mathbb{F}_2$ satisfying $P^2=P$ and $Q^3=O$.