多项式环上的二阶矩阵环$M_2\left({\mathbb{Z}}_{p^2}{}_{q^2}\left[x\right]\right)$中的幂等元

Idempotents in Matrix Rings of Order 2 over Polynomial Ring ${\mathbb{Z}}_{p^2}{}_{q^2}\left[x\right]$

摘要: 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${\mathbb{Z}}_n$ 和${\mathbb{Z}}_n\left[x\right]$ 中，可能存在非平凡幂等元。本文将研究多项式环${\mathbb{Z}}_{p^2}{}_{q^2}\left[x\right]$ 中的幂等元，并进一步探究多项式环${\mathbb{Z}}_{p^2}{}_{q^2}\left[x\right]$ 上的2阶矩阵环$M_2\left({\mathbb{Z}}_{p^2}{}_{q^2}\left[x\right]\right)$ 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，${\mathbb{Z}}_{p^2}{}_{q^2}\left[x\right]$ 中有4个幂等元，$M_2\left({\mathbb{Z}}_{p^2}{}_{q^2}\left[x\right]\right)$ 中有7个非平凡幂等矩阵。记环$R$ 的幂等元集合为$Id\left(R\right)$ 。

Abstract: Idempotents are a type of important elements in a ring. An element $a$ is idempotent, if and only if $a^2=a$ . Any ring containing a unity typically has two idempotent elements, namely 0 and 1, which are called the trivial idempotents. However, in rings such as ${\mathbb{Z}}_n$ and ${\mathbb{Z}}_n\left[x\right]$ , there may exist non-trivial idempotents. This paper studies the idempotents in the polynomial ring ${\mathbb{Z}}_{p^2}{}_{q^2}\left[x\right]$ , where $p$ , $q$ are distinct primes, and further investigates the forms and properties of non-trivial idempotents in the $2\times 2$ matrix ring $M_2\left({\mathbb{Z}}_{p^2}{}_{q^2}\left[x\right]\right)$ . The study shows that there are four idempotent elements in ${\mathbb{Z}}_{p^2}{}_{q^2}\left[x\right]$ and seven non-trivial idempotent matrices in $M_2\left({\mathbb{Z}}_{p^2}{}_{q^2}\left[x\right]\right)$ . The set of idempotents of a ring $R$ is denoted by$Id\left(R\right)$ .