论文标题
$ 2 \ times 2 $矩阵戒指上的iDempotents超越正式电源系列
Idempotents of $2\times 2$ matrix rings over rings of formal power series
论文作者
论文摘要
令$ a_1，\ ldots，a_s $为单一的交换环，没有非平凡的iDempotents，让$ a = a_1 \ oplus \ cdots \ cdots \ oplus a_s $是他们的直接总和。我们描述了$ 2 \ times 2 $矩阵环$ m_2（a [[x]）$中的所有diadempotents上的正式电力系列的$ a [[x]] $，具有$ a $ a $的系数，并在任意变量集中$ x $。我们将此结果应用于矩阵环$ M_2（{\ Mathbb z} _n [[x]）$上的$ $ a = {\ mathbb z} _n $，中文提醒定理和Euler-Fermat定理。
Let $A_1,\ldots,A_s$ be unitary commutative rings which do not have non-trivial idempotents and let $A=A_1\oplus\cdots\oplus A_s$ be their direct sum. We describe all idempotents in the $2\times 2$ matrix ring $M_2(A[[X]])$ over the ring $A[[X]]$ of formal power series with coefficients in $A$ and in arbitrary set of variables $X$. We apply this result to the matrix ring $M_2({\mathbb Z}_n[[X]])$ over the ring ${\mathbb Z}_n[[X]]$ for an arbitrary positive integer $n$ greater than 1. Our proof is elementary and uses only the Cayley-Hamilton theorem (for $2\times 2$ matrices only) and, in the special case $A={\mathbb Z}_n$, the Chinese reminder theorem and the Euler-Fermat theorem.
