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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.54 no.2 Bogotá July/Dec. 2020 Epub Mar 04, 2021
https://doi.org/10.15446/recolma.v54n2.93833
Artículos originales
Certain Properties of Square Matrices over Fields with Applications to Rings
Algunas propiedades de matrices cuadradas sobre cuerpos con aplicaciones a anillos
1 Bulgarian Academy of Sciences, Sofia, Bulgaria
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 & 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. & Astr. (2019) and Chebyshevskii Sb. (2019), respectively.
Keywords: Nilpotent matrices; idempotent matrices; Jordan canonical form; algebraically closed fields; super π-regular rings
Probamos que toda matriz cuadrada nilpotente sobre un cuerpo es igual a la resta de dos matrices idempotentes, también probamos que toda matriz cuadrada con coeficientes en un cuerpo algebraicamente cerrado es la suma de una matriz nilpotente cuyo cuadrado es nulo y una matriz diagonalizable. También aplicamos estos resultados en una variante de anillos π-regulares. Estos resultados mejoran los resultados presentados por Breaz en Linear Algebra & Appl. (2018) y aquellos de Abyzov presentados en Siberian Math. J. (2019) al igual que aquellos publicados por el autor del presente artículo en Vest. St. Petersburg Univ. - Ser. Math., Mech. & Astr. (2019) y en Chebyshevskii Sb. (2019), respectivamente.
Palabras clave: Matrices nilpotentes; matrices idempotentes; forma canónica de Jordan; cuerpos algebraicamente cerrados; anillos π-regulares
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Received: April 06, 2020; Accepted: June 05, 2020
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