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On the Regularity and Clean Properties of Matrices over Dual Numbers

  • Berrin Şentürk
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Abstract

In this paper, we study regularity and clean-type decompositions for matrices over dual numbers. We characterize the von Neumann regularity of dual matrices by establishing a necessary and sufficient compatibility condition between their real and dual components. At the element level, we determine the conditions under which an individual dual matrix ℳ = A + Bε is regular, π-regular, nil-clean, or strongly nil-clean. Using the solvability of Sylvester matrix equations, we show that the matrix ring over dual numbers Mn(𝔻) is strongly π-regular and strongly clean. Moreover, Mn(𝔻) is π-regular, clean, r-clean, strongly r-clean, NR-clean, and strongly NR-clean. On the other hand, Mn(𝔻) fails to be regular, strongly regular, nil-clean, strongly nil-clean, and uniquely NR-clean, although it inherits several strong decomposition properties from Mn(ℝ). Our characterizations at the element level show exactly when each property holds.

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Cite

BibTeXRIS
BibTeX
@misc{senturk2026regularity,
  title = {{On the Regularity and Clean Properties of Matrices over Dual Numbers}},
  author = {Berrin Şentürk},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.29468},
  eprint = {2609.29468},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29468},
}
RIS
TY  - UNPB
TI  - On the Regularity and Clean Properties of Matrices over Dual Numbers
AU  - Berrin Şentürk
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.29468
UR  - https://arxiv.org/abs/2609.29468
AB  - In this paper, we study regularity and clean-type decompositions for matrices over dual numbers. We characterize the von Neumann regularity of dual matrices by establishing a necessary and sufficient compatibility condition between their real and dual components. At the element level, we determine the conditions under which an individual dual matrix ℳ = A + Bε is regular, π-regular, nil-clean, or strongly nil-clean. Using the solvability of Sylvester matrix equations, we show that the matrix ring over dual numbers M_n(𝔻) is strongly π-regular and strongly clean. Moreover, M_n(𝔻) is π-regular, clean, r-clean, strongly r-clean, NR-clean, and strongly NR-clean. On the other hand, M_n(𝔻) fails to be regular, strongly regular, nil-clean, strongly nil-clean, and uniquely NR-clean, although it inherits several strong decomposition properties from M_n(ℝ). Our characterizations at the element level show exactly when each property holds.
ER  -