Centralizers in endomorphism rings

Original · EN

We prove that the centralizer Cen(f) in Homᵣ(M,M) of a nilpotent endomorphism f of a finitely generated semisimple left R-module 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 Mₘ(R[z]), where m is the dimension (composition length) of ker(f). If R is a local ring, then we provide an explicit description of the above Cen(f). If in addition Z(R) is a field and R/J(R) is finite dimensional over Z(R), then we give a formula for the Z(R)-dimension of Cen(f). If R is a local ring, f is as above and g is an arbitrary element of Homᵣ(M,M), then we give a complete description of the containment Cen(f) in Cen(g) in terms of an appropriate R-generating set of M. Using our results about nilpotent endomorphisms, for an arbitrary (not necessarily nilpotent) linear map f in Homₖ(V,V) of a finite dimensional vector space V over a field K we determine the PI-degree of Cen(f) and give other information about the polynomial identities of Cen(f).