Idempotents of 2 × 2 matrix rings over rings of formal power series,Linear and Multilinear Algebra

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Idempotents of 2 × 2 matrix rings over rings of formal power series
Linear and Multilinear Algebra ( IF 0.9 ) Pub Date : 2021-04-15 , DOI: 10.1080/03081087.2021.1910121
Vesselin Drensky ₁

Affiliation

ABSTRACT

Let $A_{1}, \dots, A_{s}$ be unitary commutative rings which do not have non-trivial idempotents and let $A = A_{1} \oplus \dots \oplus A_{s}$ 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 an arbitrary set of variables X. We apply this result to the matrix ring $M_{2} (Z_{n} [[X]])$ over the ring $Z_{n} [[X]]$ where $Z_{n} ≅ Z / n Z$ 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 $A = Z_{n}$ , the Chinese remainder theorem and the Euler-Fermat theorem.

更新日期：2021-04-15

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