Centralizers in endomorphism rings

Abstract

Communicated by Efim Zelmanov MSC: 15A30 15A27 16D60 16R10 16S50 16U70 Keywords: Centralizer Module endomorphism Nilpotent Jordan normal base We prove that the centralizer Cen(ϕ) ⊆ End R (M) of a nilpotent endomorphism ϕ 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 × 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 PIdegree of Cen( A).

Some ring theory from Jenő Szigeti

“…An astonishing fact is that a far reaching generalization of this fundamental result can be formulated for nilpotent complete join homomorphisms of lattices (see [13]). The description of the centralizers and zero level centralizers (two sided annihilators) in Szigeti's papers ( [1,17]) depends on the use of the following remarkable consequence of this lattice theoretical Jordan normal base theorem.…”

“…The rest of the section follows literally the exposition in [1] and [17]. If R is a local ring with Jacobson radical J and R M is a finitely generated semisimple left R-module, then the centralizer Cen.…”

The zero-level centralizer in endomorphism algebras

“…In order to provide a self-contained treatment, we collect some notation, definitions and statements from [1,6]. Let Z(R) and J = J(R) denote the centre and the Jacobson radical of a ring R (with identity).…”

“…Our treatment follows the arguments of [1] and is heavily based on the results of [1,6]. Our treatment follows the arguments of [1] and is heavily based on the results of [1,6].…”

“…In § 2 we consider a fixed nilpotent Jordan normal base of R M with respect to a given nilpotent ϕ ∈ End R (M ) and present all the necessary prerequisites from [1,6]. Section 3 is devoted entirely to the nilpotent case.…”