Certain Properties of Square Matrices over Fields with Applications to Rings

Peter V Danchev

doi:10.15446/recolma.v54n2.93833

Abstract

We prove that any square nilpotent matrix over a field is a difference of two idempotent matrices as well as that any square matrix over an algebraically closed field is a sum of a nilpotent square-zero matrix and a diagonalizable matrix. We further apply these two assertions to a variation of π-regular rings. These results somewhat improve on establishments due to Breaz from Linear Algebra & amp; Appl. (2018) and Abyzov from Siberian Math. J. (2019) as well as they also refine two recent achievements due to the present author, published in Vest. St. Petersburg Univ. - Ser. Math., Mech. & amp; Astr. (2019) and Chebyshevskii Sb. (2019), respectively.

Highlights

Introduction and FundamentalsAll rings R are assumed here to be associative, containing the identity element 1 which differs from the zero element 0 of R
Recall that a ring R is said to be π-regular, provided every element r is π-regular, that is, there exists n ∈ N depending on r such that rn ∈ rnRrn, and we call a ring R super π-regular, provided that rm ∈ rmRrm for each m ∈ N – it is pretty clear that super πregularity implies π-regularity, an irreversible implication
An important class of super π-regular rings is the class of von Neumann regular rings in the sense that they are π-regular with n = 1 for all r

Summary

Introduction

Introduction and FundamentalsAll rings R are assumed here to be associative, containing the identity element 1 which differs from the zero element 0 of R. The presentation of a matrix over a ring as a sum/difference of some special elements like units, nilpotents, idempotents, potents, etc., always plays a central role in matrix ring theory.

Results

Conclusion