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Pure Mathematics 2026

多项式环上的二阶矩阵环 $M_{2} (？_{p^{2}}_{q^{2}} [x])$ 中的幂等元
Idempotents in Matrix Rings of Order 2 Over Polynomial Ring $？_{p^{2}}_{q^{2}} [x]$

DOI: 10.12677/pm.2026.161013, PP. 105-111

张蓉, 蒋博俊

Keywords: 幂等元，矩阵环，多项式环
Idempotent, Matrix Ring, Polynomial Ring

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Abstract:

在环论中，幂等元是很重要的一类元素，幂等元是指满足 $a^{2} = a$ 的元素 $a$ 。任何含单位元的环通常都有两个幂等元，即0和1，这两个特殊的幂等元通常被称为平凡幂等元。然而，在环 $？_{n}$ 和 $？_{n} [x]$ 中，可能存在非平凡幂等元。本文将研究多项式环 $？_{p^{2}}_{q^{2}} [x]$ 中的幂等元，并进一步探究多项式环 $？_{p^{2}}_{q^{2}} [x]$ 上的2阶矩阵环 $M_{2} (？_{p^{2}}_{q^{2}} [x])$ 中非平凡幂等元的形式与性质，其中 $p$ 、 $q$ 为不同素数。研究结果表明， $？_{p^{2}}_{q^{2}} [x$

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多项式环上的二阶矩阵环 M 2 ( ？ p 2 q 2 [ x ] )中的幂等元Idempotents in Matrix Rings of Order 2 Over Polynomial Ring ？ p 2 q 2 [ x ]

多项式环上的二阶矩阵环 $M_{2} (？_{p^{2}}_{q^{2}} [x])$ 中的幂等元
Idempotents in Matrix Rings of Order 2 Over Polynomial Ring $？_{p^{2}}_{q^{2}} [x]$