Centralizers in endomorphism rings

We prove that the centralizer Cen(phi) subset of End(R)(M) of a nilpotent endomorphism phi of a finitely generated semisimple left R module (R)M (over an arbitrary ring R) is the homomorphic Image of the opposite of a certain Z(R)-subalgebra of the full m x m matrix algebra M(m)(R[z]) where m is the dimension of ker(phi) If R is a local ring then we give a complete characterization of Cen(phi) and of the containment Cen(phi) subset of Cen(sigma) where a 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) (C) 2010 Elsevier Inc All rights reserved