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

If $R$ is a commutative unital ring and $M$ is a unital $R$-module, then each element of ${\mathrm{End}}_R(M)$ determines a left ${\mathrm{End}}_R(M)[X]$-module structure on ${\mathrm{End}}_R(M)$, where ${\mathrm{End}}_R(M)$ is the $R$-algebra of endomorphisms of $M$ and ${\mathrm{End}}_R(M)[X]={\mathrm{End}}_R(M){\otimes }_{R}R[X]$. 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.