Unit and idempotent additive maps over countable linear transformations

Year 2024, Volume: 53 Issue: 2, 305 - 313, 23.04.2024

Günseli Gümüşel , M. Tamer Koşan , Jan Zemlıcka

https://doi.org/10.15672/hujms.1187608

Abstract

Let $V$ be a countably generated right vector space over a field $F$ and $\sigma\in End(V_F)$ be a shift operator. We show that there exist a unit $u$ and an idempotent $e$ in $End(V_F)$ such that $1-u,\sigma-u$ are units in $End(V_F)$ and $1-e,\sigma-e$ are idempotents in $End(V_F)$. We also obtain that if $D$ is a division ring $D\ncong \mathbb Z_2, \mathbb Z_3 $ and $V_D$ is a $D$-module, then for every $\alpha\in End(V_D)$ there exists a unit $u\in End(V_D)$ such that $1-u,\alpha-u$ are units in $End(V_D)$.

Keywords

unit, shift operator, idempotent matrix, tripotent matrix, semilocal ring, division ring

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