关于对偶数矩阵的正则性与洁净性质
On the Regularity and Clean Properties of Matrices over Dual Numbers
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中文总结 AI 辅助
本文研究对偶数矩阵的正则性与洁净分解，通过相容性条件刻画冯·诺依曼正则性，并证明矩阵环 $M_n(\mathbb{D})$ 是强 $\pi$-正则且强洁净的，同时指出其非正则、非幂零洁净等性质。
中文摘要 AI 辅助
本文研究了对偶数矩阵的正则性与洁净型分解。我们通过建立对偶矩阵的实部与对偶部之间的必要且充分的相容性条件，刻画了其对偶矩阵的冯·诺依曼正则性。我们确定了双矩阵 $\mathcal{M} = A + B\epsilon$ 为正则、$\pi$-正则、幂零洁净或强幂零洁净的条件。利用西尔维斯特矩阵方程的可解性，我们证明了对偶数上的矩阵环 $M_n(\mathbb{D})$ 是强 $\pi$-正则且强洁净的。此外，$M_n(\mathbb{D})$ 是 $\pi$-正则、洁净、$r$-洁净、强 $r$-洁净、$\mathrm{NR}$-洁净和强 $\mathrm{NR}$-洁净的。另一方面，$M_n(\mathbb{D})$ 不是正则、强正则、幂零洁净、强幂零洁净和唯一 $\mathrm{NR}$-洁净的，尽管它从 $M_n(\mathbb{R})$ 继承了若干强分解性质。
英文摘要
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. We determine the conditions under which a 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 $M_n(\mathbb{R})$.
