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

TL;DR

The paper characterizes all idempotent elements in the ring of 2x2 matrices over formal power series with coefficients in certain rings, providing explicit descriptions and applications to rings over integers modulo n.

Contribution

It provides a complete description of idempotents in 2x2 matrix rings over formal power series rings for rings without non-trivial idempotents, using elementary methods.

Findings

Explicit description of idempotents in M_2(A[[X]])

Application to rings over Z_n[[X]] for any n>1

Elementary proof using Cayley-Hamilton, Chinese remainder, and Euler-Fermat theorems

Abstract

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.