The centralizer of an element in an endomorphism ring

We prove that the centralizer Cφ ⊆ HomR(M,M) of a nilpotent endomorphism φ of a finitely generated semisimple left R-module RM (over an arbitrary ring R) is the homomorphic image of the opposite of a certain Z(R)-subalgebra of the full m×m matrix algebra Mm×m(R[t]), where m is the dimension (composition length) of ker(φ). If R is a finite dimensional division ring over its central subfield Z(R) and φ is nilpotent, then we give an upper bound for the Z(R)-dimension of Cφ. If R is a local ring, φ is nilpotent and σ ∈HomR(M,M) is arbitrary, then we provide a complete description of the containment Cφ ⊆ Cσ in terms of an appropriate R-generating set of RM . For an arbitrary (not necessarily nilpotent) linear map φ ∈HomK(V, V ) of a finite dimensional vector space V over an algebraically closed field K we prove that Cφ is the homomorphic image of a direct product of p factors such that for each 1 ≤ i ≤ p the i-th factor is a K-subalgebra of Mmi×mi(K[t]) with mi = dim(ker(φ − λi1V )) and {λ1, λ2, . . . , λp} is the set of all eigenvalues of φ. As a consequence, we obtain that Cφ satisfies all polynomial identities of Mm×m(K[t]), where m is the maximum of the mi’s.