Abstract
If R is a commutative unital ring and M is a unital R-module, then each element of EndR(M) determines a left EndR(M)[X]-module structure on EndR(M), where EndR(M) is the R-algebra of endomorphisms of M and EndR(M)[X]=EndR(M)⊗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.
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