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Unit and idempotent additive maps over countable linear transformations

Yıl 2024, Cilt: 53 Sayı: 2, 305 - 313, 23.04.2024
https://doi.org/10.15672/hujms.1187608

Öz

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 $End(V_D)$.

Kaynakça

  • [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.
Yıl 2024, Cilt: 53 Sayı: 2, 305 - 313, 23.04.2024
https://doi.org/10.15672/hujms.1187608

Öz

Kaynakça

  • [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.
Toplam 13 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Günseli Gümüşel 0000-0001-8068-4294

M. Tamer Koşan 0000-0003-1775-2957

Jan Zemlıcka 0000-0003-3319-5193

Erken Görünüm Tarihi 10 Ocak 2024
Yayımlanma Tarihi 23 Nisan 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 53 Sayı: 2

Kaynak Göster

APA Gümüşel, G., Koşan, M. . T., & Zemlıcka, J. (2024). Unit and idempotent additive maps over countable linear transformations. Hacettepe Journal of Mathematics and Statistics, 53(2), 305-313. https://doi.org/10.15672/hujms.1187608
AMA Gümüşel G, Koşan MT, Zemlıcka J. Unit and idempotent additive maps over countable linear transformations. Hacettepe Journal of Mathematics and Statistics. Nisan 2024;53(2):305-313. doi:10.15672/hujms.1187608
Chicago Gümüşel, Günseli, M. Tamer Koşan, ve Jan Zemlıcka. “Unit and Idempotent Additive Maps over Countable Linear Transformations”. Hacettepe Journal of Mathematics and Statistics 53, sy. 2 (Nisan 2024): 305-13. https://doi.org/10.15672/hujms.1187608.
EndNote Gümüşel G, Koşan MT, Zemlıcka J (01 Nisan 2024) Unit and idempotent additive maps over countable linear transformations. Hacettepe Journal of Mathematics and Statistics 53 2 305–313.
IEEE G. Gümüşel, M. . T. Koşan, ve J. Zemlıcka, “Unit and idempotent additive maps over countable linear transformations”, Hacettepe Journal of Mathematics and Statistics, c. 53, sy. 2, ss. 305–313, 2024, doi: 10.15672/hujms.1187608.
ISNAD Gümüşel, Günseli vd. “Unit and Idempotent Additive Maps over Countable Linear Transformations”. Hacettepe Journal of Mathematics and Statistics 53/2 (Nisan 2024), 305-313. https://doi.org/10.15672/hujms.1187608.
JAMA Gümüşel G, Koşan MT, Zemlıcka J. Unit and idempotent additive maps over countable linear transformations. Hacettepe Journal of Mathematics and Statistics. 2024;53:305–313.
MLA Gümüşel, Günseli vd. “Unit and Idempotent Additive Maps over Countable Linear Transformations”. Hacettepe Journal of Mathematics and Statistics, c. 53, sy. 2, 2024, ss. 305-13, doi:10.15672/hujms.1187608.
Vancouver Gümüşel G, Koşan MT, Zemlıcka J. Unit and idempotent additive maps over countable linear transformations. Hacettepe Journal of Mathematics and Statistics. 2024;53(2):305-13.