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

https://izlik.org/JA66GJ99KD

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

References

• [1] V.P. Camillo and J. J. Simon, The Nicholson-Varadarajan Theorem on clean linear transformations, Glasg. Math. J. 44, 365369, 2002.
• [2] H. Chen, Decompositions of countable linear transformations, Glasg. Math. J. 52 (3), 427433, 2010.
• [3] H. Chen, Decompositions of linear Transformations over division rings, Algebra Colloq. 19 (3), 459-464, 2012.
• [4] B. Goldsmith, S. Pabst and A. Scott, Unit sum numbers of rings and modules, Q. J. Math. 49 (3), 331-344, 1998.
• [5] K.R. Goodearl and P. Menal, Stable range one for rings with many units, J. Pure Appl. Algebra 54, 261-287, 1998.
• [6] M.T. Kosan, S. Sahinkaya and Y. Zhou, Additive maps on units of rings, Canad. Math. Bull. 61 (1), 130-141, 2018.
• [7] M.T. Kosan and Y. Zhou, A class of rings with the 2-sum property, Appl. Algebra Engrg. Comm. Comput. 32 (3), 399-408, 2021.
• [8] C. Li, L. Wang and Y. Zhou, On rings with the Goodearl-Menal condition, Comm. Algebra 40 (12), 4679-4692, 2012.
• [9] W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229, 269-278, 1977.
• [10] W.K. Nicholson, K. Varadarajan, Countable linear transformations are clean, Proc. Amer. Math. Soc. 126 (1), 6164, 1998.
• [11] L. Wang and Y. Zhou, Decomposing linear transformations, Bull. Aust. Math. Soc. 83, 256261, 2011.
• [12] K.G. Wolfson, An ideal-theoretic characterization of the ring of all linear transformations, Amer. J. Math. 75, 358-386, 1953.
• [13] D. Zelinsky, Every linear transformation is sum of nonsingular ones, Proc. Amer. Math. Soc. 5, 627-630, 1954.