Algebra

2510 Submissions

[2] viXra:2510.0145 [pdf] submitted on 2025-10-29 15:24:47

Four Squares Represented as Sum of Three Out of Four Variables

Authors: Oliver Couto
Comments: 4 Pages.

In volume 2 of L. Dickson book (history of theory of numbers) the below system of equation, has been considered in. Ajai Choudry in his latest paper (ref # 1), has considered three simultaneous equation with one of them having a maximum degree of three. This paper has considered a system of four equations with a maximum degree of two. The author has provided two methods to arrive at parameterization.
Category: Algebra

[1] viXra:2510.0117 [pdf] submitted on 2025-10-24 21:00:39

Tripotent-Invertible Decomposition of Matrices Over Local Rings

Authors: Huanyin Chen, Marjan Sheibani
Comments: 13 Pages.

We establish the first necessary and sufficient conditions for a $2times 2$ matrix over a local ring to be decomposable into the sum of a tripotent and an invertible matrix. Building on this decomposition, we derive a novel characterization of the generalized Drazin inverse for such matrices. Our approach hinges on a key relationship between this decomposition and the associated tripotent-quasinilpotent decomposition, thereby offering significant new insights into the spectral theory of matrices over local rings.
Category: Algebra