k[x]-modules and Core-Nilpotent endomorphisms

TL;DR

This paper characterizes core-nilpotent endomorphisms on arbitrary vector spaces via their $k[x]$-module structure, introducing a generalized inverse and decomposition extending classical matrix results.

Contribution

It provides a new description of core-nilpotent endomorphisms using $k[x]$-modules and introduces a generalized inverse and decomposition applicable to arbitrary vector spaces.

Findings

Describes core-nilpotent endomorphisms through $k[x]$-module structure.

Introduces a generalized inverse extending the Drazin inverse.

Presents a generalized core-nilpotent decomposition for arbitrary vector spaces.

Abstract

Core-nilpotent endomorphisms over an arbitrary vector space form the largest subset of the ring of endomorphisms over that arbitrary vector space which admit a decomposition as sum of two endomorphisms satisfying the analogous properties as the well known core-nilpotent decomposition of matrices. In this paper we present a new description of core-nilpotent endomorphisms using the $k[x]-$module structure they define in the base vector space. Moreover, our approach provides us with a ``new'' generalized inverse that restricts to the well known Drazin inverse under certain conditions. Similarly, we present a generalized core-nilpotent decomposition for endomorphisms over arbitrary vector spaces.