Unit and idempotent additive maps over countable linear transformations

Authors : Günseli Gümüşel, M Tamer Koşan, Jan Zemlıcka

Pages : 305-313

Doi:10.15672/hujms.1187608

View : 430 | Download : 432

Publication Date : 2024-04-23

Article Type : Research Paper

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