Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Certain Clean Decompositions for Matrices over Local Rings

  • Yosum Kurtulmaz (Department of Mathematics, Bilkent University) ;
  • Handan Kose (Department of Mathematics, Ahi Evran University) ;
  • Huanyin Chen (School of Big Data, Fuzhou University of International Studies and Trade)
  • Received : 2022.12.15
  • Accepted : 2023.07.25
  • Published : 2023.12.31
Download PDF
⟨ Previous Next ⟩

Abstract

An element a ∈ R is strongly rad-clean provided that there exists an idempotent e ∈ R such that a - e ∈ U(R), ae = ea and eae ∈ J(eRe). In this article, we completely determine when a 2 × 2 matrix over a commutative local ring is strongly rad clean. An application to matrices over power-series is also given.

Keywords

References

  1. D. D. Anderson and V. P. Camillo, Commutative rings whose elements are a sum of a unit and idempotent, Comm. Algebra, 30(7)(2002), 3327-3336. https://doi.org/10.1081/AGB-120004490
  2. G. Borooah, A. J. Diesl and T. J. Dorsey, Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra, 212(1)(2008), 281-296. https://doi.org/10.1016/j.jpaa.2007.05.020
  3. S. Breaz, Matrices over finite fields as sums of periodic and nilpotent elements, Linear Algebra Appl., 555(2018), 92-97. https://doi.org/10.1016/j.laa.2018.06.017
  4. H. Chen, Rings Related to Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011.
  5. J. Cui and J. Chen, When is a 2×2 matrix ring over a commutative local ring quasipolar?, Comm. Algebra, 39(9)(2011), 3212-3221. https://doi.org/10.1080/00927872.2010.499118
  6. P. Danchev, Certain properties of square matrices over fields with applications to rings, Rev. Colombiana Mat., 54(2)(2020), 109-116. https://doi.org/10.15446/recolma.v54n2.93833
  7. P. Danchev, E. Garcia and M. G. Lozano, Decompositions of matrices into potent and square-zero matrices, Internat. J. Algebra Comput., 32(2)(2022), 251-263. https://doi.org/10.1142/S0218196722500126
  8. A. J. Diesl and T. J. Dorsey, Strongly clean matrices over arbitrary rings, J. Algebra, 399(2014), 854-869. https://doi.org/10.1016/j.jalgebra.2013.08.044
  9. Y. Li, Strongly clean matrix rings over local rings, J. Algebra, 312(1)(2007), 397-404. https://doi.org/10.1016/j.jalgebra.2006.10.032
  10. J. Ster, On expressing matrices over ℤ2 as the sum of an idempotent and a nilpotent, Linear Algebra Appl., 544(10)(2018), 339-349. https://doi.org/10.1016/j.laa.2018.01.015
  11. G. Tang, Y. Zhou, H. Su, Matrices over a commutative ring as sums of three idempotents or three involutions, Linear Multilinear Algebra, 67(2)(2019), 267-2