Proof of Cayley-Hamilton theorem using polynomials over the algebra of module endomorphisms

TL;DR

This paper presents a concise proof of the Cayley-Hamilton theorem using polynomial modules over the algebra of module endomorphisms, offering a reformulation and generalizations of the classical result.

Contribution

It introduces a novel approach to prove the Cayley-Hamilton theorem via module structures over endomorphism algebras, simplifying and extending existing proofs.

Findings

Provides a short proof of Cayley-Hamilton theorem

Reformulates the proof using module structures over endomorphism algebras

Extends to generalizations of the classical theorem

Abstract

If $R$ is a commutative unital ring and $M$ is a unital $R$-module, then each element of $\operatorname{End}_R(M)$ determines a left $\operatorname{End}_{R}(M)[X]$-module structure on $\operatorname{End}_{R}(M)$, where $\operatorname{End}_{R}(M)$ is the $R$-algebra of endomorphisms of $M$ and $\operatorname{End}_{R}(M)[X] =\operatorname{End}_{R}(M)\otimes_RR[X]$. These structures provide a very short proof of the Cayley-Hamilton theorem, which may be viewed as a reformulation of the proof in Algebra by Serge Lang. Some generalisations of the Cayley-Hamilton theorem can be easily proved using the proposed method.