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Article Dans Une Revue Linear Algebra and its Applications Année : 2022

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

Alexey Muranov

Résumé

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.

Dates et versions

hal-03230521 , version 1 (20-05-2021)

Identifiants

Citer

Alexey Muranov. Proof of Cayley-Hamilton theorem using polynomials over the algebra of module endomorphisms. Linear Algebra and its Applications, 2022, 645, pp.165-169. ⟨10.1016/j.laa.2022.03.012⟩. ⟨hal-03230521⟩
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