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arXiv · 2609.29468

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 $\mathcal{M} = A + Bε$ 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 $M_n(\mathbb{D})$ is strongly $π$-regular and strongly clean. Moreover, $M_n(\mathbb{D})$ is $π$-regular, clean, $r$-clean, strongly $r$-clean, $\mathrm{NR}$-clean, and strongly $\mathrm{NR}$-clean. On the other hand, $M_n(\mathbb{D})$ fails to be regular, strongly regular, nil-clean, strongly nil-clean, and uniquely $\mathrm{NR}$-clean, although it inherits several strong decomposition properties from $M_n(\mathbb{R})$. Our characterizations at the element level show exactly when each property holds.

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Berrin Şentürk. 2026-10-19. On the Regularity and Clean Properties of Matrices over Dual Numbers. https://arxiv.org/abs/2609.29468

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