Strongly Clean Matrices over Commutative Rings

TL;DR

This paper characterizes when matrices over projective-free commutative rings are strongly clean, linking this property to the factorization of their characteristic polynomials and extending previous results to power series rings.

Contribution

It provides a new characterization of strongly clean matrices over projective-free rings via polynomial factorizations, extending existing theorems to broader classes of rings.

Findings

Matrices with a given characteristic polynomial are strongly clean iff the companion matrix is strongly clean.

Strongly clean property is equivalent to specific polynomial factorizations.

Results extend to matrices over power series rings over projective 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].