Centralizers in endomorphism rings

Abstract

We prove that the centralizer Cen ( φ ) ⊆ End R ( M ) of a nilpotent endomorphism φ of a finitely generated semisimple left R-module M R (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 M m ( 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 End R ( 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 ) .