Decompositions of Matrices into Potent and Square-Zero Matrices

TL;DR

This paper proves that any square matrix over a finite commutative ring with a Jacobson radical of zero-square can be expressed as a sum of a potent and a nilpotent matrix of order at most two, extending previous results.

Contribution

It generalizes matrix decomposition results to broader rings, improving upon earlier theorems for matrices over fields and specific rings.

Findings

Every matrix over the specified ring is representable as a sum of a potent and a nilpotent matrix.

The nilpotent component has order at most two.

The result extends previous theorems to more general ring settings.

Abstract

In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such a matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most two when the Jacobson radical of the ring has zero-square. This somewhat extends results of ours in Lin. & Multilin. Algebra (2021) established for matrices considered on arbitrary fields. Our main theorem also improves on recent results due to Abyzov et al. in Mat. Zametki (2017), \v{S}ter in Lin. Algebra & Appl. (2018) and Shitov in Indag. Math. (2019).