有限局部环上幂零矩阵与幂等矩阵的乘积
Products of Nilpotent and Idempotent Matrices over Finite Local Rings
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中文总结 AI 辅助
本文研究有限交换局部主环R上的2阶矩阵，证明幂零与幂等矩阵的乘积必为幂等或幂零矩阵的乘积，IN-与NI-矩阵类重合，还给出其元素数量的显式公式。
中文摘要 AI 辅助
设R为有限交换局部主环，本文研究M₂(R)中幂零矩阵与幂等矩阵的乘积，证明每一个幂零矩阵与幂等矩阵的乘积要么是幂等矩阵的乘积，要么是幂零矩阵的乘积。接着考虑IN-矩阵与NI-矩阵，即分别可写成幂等矩阵乘幂零矩阵、幂零矩阵乘幂等矩阵的矩阵，证明M₂(R)中IN-矩阵类与NI-矩阵类重合，并给出它们共同类的显式描述。最后，当|R|=qⁿ且R/J(R)≅GF(q)时，证明|IN(M₂(R))|=|NI(M₂(R))|=q^(3n-2)(qⁿ+q²-1)。
英文摘要
Let $R$ be a finite commutative local principal ring. We study products of nilpotent and idempotent matrices in $M_2(R)$. We show that every product of nilpotent and idempotent matrices is either a product of idempotents or a product of nilpotents. We then consider IN- and NI-matrices, that is, matrices which can be written as a product of an idempotent and a nilpotent matrix in the respective orders. We prove that the classes of IN- and NI-matrices in $M_2(R)$ coincide and give an explicit description of their common class. Finally, if $|R|=q^n$ and $R/J(R)\cong GF(q)$, we show that \[ |\operatorname{IN}(M_2(R))| = |\operatorname{NI}(M_2(R))| = q^{3n-2}(q^n+q^2-1). \]
