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