Inglês (pdf)
Artigo em XML
Referências do artigo
Tradução automática
Enviar este artigo por email
Permalink
Texto PDF
Serviços Personalizados
Journal
Artigo
Indicadores
-
Citado por SciELO -
Acessos
Links relacionados
-
Citado por Google -
Similares em
SciELO -
Similares em Google
Compartilhar
Revista Colombiana de Matemáticas
versão impressa ISSN 0034-7426
Rev.colomb.mat. vol.54 no.2 Bogotá jul./dez. 2020 Epub 04-Mar-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
References
[1]. A. N. Abyzov, Strongly q-nil-clean rings, Siber. Math. J. 60 (2019), no. 2, 197-208. [ Links ]
[2]. A. N. Abyzov and I. I. Mukhametgaliev, On some matrix analogs of the little Fermat theorem, Math. Notes 101 (2017), no. 1-2, 187-192. [ Links ]
[3]. K. I. Beidar, K. C. O'Meara, and R. M. Raphael, On uniform diagonalisation of matrices over regular rings and one-accesible regular algebras, Commun. Algebra 32 (2004), 3543-3562. [ Links ]
[4]. W. Boucher and F. Ulmer, Linear codes using skew polynomials with automorphisms and derivations, Designs, Codes and Cryptography, Springer Verlag, 70, 2014, 405-431. [ Links ]
[5]. S. Breaz, Matrices over finite fields as sums of periodic and nilpotent elements, Lin. Alg. & Appl. 555 (2018), 92-97. [ Links ]
[6]. S. Breaz, G. Câlugâreanu, P. Danchev, and T. Micu, Nil-clean matrix rings, Lin. Alg. & Appl. 439 (2013), 3115-3119. [ Links ]
[7]. M. P. Cuéllar, J. Gómez-Torrecillas, F. J. Lobillo, and G. G. Navarro, Genetic algorithms with permutation-based representation for computing the distance of linear codes, arXiv:2002.12330v1, arXiv:1810.01260. [ Links ]
[8]. P. Danchev, E. García, and M. G. Lozano, Decompositions of matrices into diagonalizable and square-zero matrices, Lin. & Multilin. Algebra 69 (2021). [ Links ]
[9]. P. V. Danchev, A generalization of (-regular rings, Turk. J. Math. 43(2019), 702-711. [ Links ]
[10] ______, On a property of nilpotent matrices over an algebraically closed field, Chebyshevskii Sbornik 20 (2019), no. 3, 400-403. [ Links ]
[11] ______, Weakly exchange rings whose units are sums of two idempotents, Vestnik of St. Petersburg Univ, Ser. Math., Mech. & Astr. 6(64) (2019), no. 2, 265-269. [ Links ]
[12] ______, Representing matrices over fields as square-zero matrices and diagonal matrices, Chebyshevskii Sbornik 21 (2020), no. 3. [ Links ]
[13]. E. García, M. G. Lozano, R. M. Alcázar, and G. Vera de Salas, A Jordan canonical form for nilpotent elements in an arbitrary ring, Lin. Alg. & Appl. 581 (2019), 324-335. [ Links ]
[14]. H. Gluesing-Luerssen, Introduction to skew-polynomial rings and skew-cyclic codes, arXiv:1902.03516v2. [ Links ]
[15]. J. Gómez-Torrecillas , F. J. Lobillo , and G. Navarro, Convolutional codes with a matrix-algebra wordambient, Advances in Mathematics of Communications 10 (2016), 29-43. [ Links ]
[16] ______, A new perspective of cyclicity in convolutional codes, IEEE Transactions on Information Theory 62 (2016), 2702-2706. [ Links ]
[17] ______, Ideal codes over separable ring extensions, IEEE Transactions on Information Theory 63 (2017), 2796-2813. [ Links ]
[18]. R. E. Hartwig and M. S. Putcha, When is a matrix a difference of two idempotents, Lin. & Multilin. Algebra 26 (1990), no. 4, 267-277. [ Links ]
[19]. D. A. Jaume and R. Sota, On the core-nilpotent decomposition of trees, Lin. Alg. & Appl. 563 (2019), 207-214. [ Links ]
[20]. O. Lezama, Coding theory over noncommutative rings of polynomial type, preprint (2020). [ Links ]
[21]. K. C. O'Meara, Nilpotents often the difference of two idempotents, unpublished draft privately circulated on March 2018. [ Links ]
[22]. Y. Shitov, The ring M8k+4(Z2) is nil-clean of index four, Indag. Math. 30 (2019), 1077-1078. [ Links ]
[23]. J. Šter, On expressing matrices over Z 2 as the sum of an idempotent and a nilpotent, Lin. Alg. & Appl. 544 (2018), 339-349. [ Links ]
Received: April 06, 2020; Accepted: June 05, 2020
This is an open-access article distributed under the terms of the Creative Commons Attribution License
