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 $End(V_D)$.
Primary Language | English |
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Subjects | Mathematical Sciences |
Journal Section | Mathematics |
Authors | |
Early Pub Date | January 10, 2024 |
Publication Date | April 23, 2024 |
Published in Issue | Year 2024 Volume: 53 Issue: 2 |