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.
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Funding
The work was supported by the Russian Science Foundation and the Cabinet of Ministers of the Republic of Tatarstan (project no. 23-21-10086) and performed under the development program of Volga Region Mathematical Center (agreement no. 0075-02-2023-944).
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Brief communication presented by S.M. Skryabin
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Abyzov, A.N., Tapkin, D.T. Rings, Matrices over Which Are Representable As the Sum of Two Potent Matrices. Russ Math. 67, 82–85 (2023). https://doi.org/10.3103/S1066369X23120022
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DOI: https://doi.org/10.3103/S1066369X23120022