Centralizers in endomorphism rings

We prove that the centralizer Cen(φ)⊆ EndR(M) of a nilpotent endomorphism φ of a finitely generated semisimple left R-module (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(R[z]), where m is the dimension of ker(φ). If R is a local ring, then we give a complete characterization of Cen(φ) and of the containment Cen(φ)⊆Cen(σ), where σ is a not necessarily nilpotent element of EndR(M). For a K-linear map A of a finite dimensional vector space over a field K we determine the PI-degree of Cen(A).