Matrices that are integral over the base ring

Abstract

We study matrices over arbitrary rings that are integral over the base ring. For any ring [Formula: see text], let [Formula: see text] with [Formula: see text]. We prove that for any pairwise commuting elements [Formula: see text] if [Formula: see text], then [Formula: see text]. As a corollary, it follows that for [Formula: see text], [Formula: see text] commutative ring, if [Formula: see text] are pairwise commuting matrices such that [Formula: see text], then [Formula: see text] where [Formula: see text]. This result, which is a generalization of the Cayley–Hamilton Theorem, was proved by Phillips in 1919. For a positive integer [Formula: see text], we prove that if every matrix in [Formula: see text] satisfies a monic polynomial of degree [Formula: see text] over [Formula: see text], then [Formula: see text] is commutative. On the other hand, every diagonal matrix in [Formula: see text], [Formula: see text], satisfies a monic polynomial of degree [Formula: see text] over [Formula: see text] precisely when [Formula: see text] is a left duo ring. We prove that if every diagonal matrix in [Formula: see text], [Formula: see text], is [Formula: see text]-integral, then [Formula: see text] is Dedekind finite.