Strongly Clean Matrices over Commutative Rings

Abstract

A commutative ring $R$ is projective free provided that every finitely generated $R$-module is free. An element in a ring is strongly clean provided that it is the sum of an idempotent and a unit that commutates. Let $R$ be a projective-free ring, and let $h\in R[t]$ be a monic polynomial of degree $n$. We prove, in this article, that every $\varphi\in M_n(R)$ with characteristic polynomial $h$ is strongly clean, if and only if the companion matrix $C_h$ of $h$ is strongly clean, if and only if there exists a factorization $h=h_0h_1$ such that $h_0\in {\Bbb S}_0, h_1\in {\Bbb S}_1$ and $(h_0,h_1)=1$. Matrices over power series over projective rings are also discussed. These extend the known results [1, Theorem 12] and [5, Theorem 25].