Linear algebra in lattices and nilpotent endomorphisms of semisimple modules

Abstract

Some classical linear algebra results are translated to the language of lattices. In particular, we formulate a Jordan normal base theorem for nilpotent join homomorphisms. As an application, we obtain the existence of a Jordan normal base in a semisimple module with respect to a given nilpotent endomorphism.

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.…”

Isomorphisms between strongly triangular matrix rings

“…For more information and detailed treatment of triangular matrix rings and their applications in other areas of mathematics we refer to [3], and for some interesting related questions on matrix rings we refer to [7].…”

Centralizers in endomorphism rings

“…Since all known results about matrix centralizers are closely connected with the Jordan normal form, it is not surprising that our development depends on the existence of the nilpotent Jordan normal base of a semisimple module with respect to a given nilpotent endomorphism (guaranteed by one of the main theorems of [11]).…”