Centralizers in endomorphism rings

TL;DR

This paper characterizes the structure of centralizers of nilpotent endomorphisms in finitely generated semisimple modules over arbitrary rings, providing explicit descriptions and formulas under various conditions.

Contribution

It offers a novel algebraic description of centralizers in endomorphism rings, including explicit formulas and containment criteria, extending known results to more general rings.

Findings

Centralizer Cen(f) is a homomorphic image of a Z(R)-subalgebra of M_m(R[z])

Explicit description of Cen(f) when R is a local ring

Formula for the Z(R)-dimension of Cen(f) under certain conditions

Abstract

We prove that the centralizer Cen(f) in Hom_R(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 M_m(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 Hom_R(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 Hom_K(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).