Abstract

This paper investigates conditions under which representability of each element \(a\) from the field \(P\) as the sum \(a = f + g\) , where \({{f}^{{{{q}_{1}}}}} = f\) , \({{g}^{{{{q}_{2}}}}} = g\) , and \({{q}_{1}},{{q}_{2}}\) are fixed natural numbers >1, implies a similar representability of each square matrix over the field \(P\) . We propose a general approach to solving this problem. As an application we describe fields and commutative rings where 2 is a unit, over which each square matrix is the sum of two 4-potent matrices.

中文翻译：

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环、矩阵，可表示为两个有效矩阵的和

摘要

本文研究了场\(P\)中每个元素\(a\)可表示为和\(a = f + g\)的条件，其中\({{f}^{{{{q} _{1}}}}} = f\)、\({{g}^{{{{q}_{2}}}}} = g\)和\({{q}_{1} },{{q}_{2}}\)是固定的自然数 >1，意味着每个方阵在域\(P\)上具有相似的可表示性。我们提出了解决这个问题的通用方法。作为一个应用，我们描述域和交换环，其中 2 是一个单位，其中每个方阵都是两个 4 势矩阵的和。