Idempotents of $2\times 2$ matrix rings over rings of formal power series

Abstract

Let $A_1,\dots ,A_s$ 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.

6 pages, LATEX

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