# Strongly Clean Matrices Over Power Series

Chen, Huanyin;Kose, Handan;Kurtulmaz, Yosum

• Accepted : 2016.03.11
• Published : 2016.06.23
• 12 5

#### Abstract

An $n{\times}n$ matrix A over a commutative ring is strongly clean provided that it can be written as the sum of an idempotent matrix and an invertible matrix that commute. Let R be an arbitrary commutative ring, and let $A(x){\in}M_n(R[[x]])$. We prove, in this note, that $A(x){\in}M_n(R[[x]])$ is strongly clean if and only if $A(0){\in}M_n(R)$ is strongly clean. Strongly clean matrices over quotient rings of power series are also determined.

#### Keywords

strongly clean matrix;characteristic polynomial;power series

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