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 End R (M) determines a left End R (M) [ X ] -module structure on End R (M) , where End R (M) is the R -algebra of endomorphisms of M and End R (M) [ X ] = End R (M) ⊗ 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. [ABSTRACT FROM AUTHOR]