Article in volume
Authors:
Title:
Super strongly clean group rings
PDFSource:
Discussiones Mathematicae - General Algebra and Applications 43(1) (2023) 135-140
Received: 2021-07-29 , Revised: 2021-10-27 , Accepted: 2021-11-07 , Available online: 2023-01-12 , https://doi.org/10.7151/dmgaa.1421
Abstract:
In this paper, we study super strongly clean group rings for different classes
of rings and groups. Mainly, we prove the following results:
- Let $R$ be a ring with $2\in J(R)$ and $G$ be a locally finite $2$-group. Then the group ring $RG$ is super strongly clean if and only if $R$ is super strongly clean.
- If $R$ is a local ring with $p\in J(R)$ and $G$ is a locally finite $p$-group, then the group ring $RG$ is super strongly clean.
- If $R$ is an abelian exchange ring with $2\in J(R)$ and $G$ is a locally finite $2$-group, then the group ring $RG$ is super strongly clean.
Keywords:
super strongly clean ring, clean ring, group ring, locally finite p-group
References:
- J. Chen, W.K. Nicholson and Y. Zhou, Group rings in which every element is uniquely the sum of a unit and an idempotent, J. Algebra 306 (2006) 453–460.
https://doi.org/10.1016/j.jalgebra.2006.08.012 - J. Chen and Y. Zhou, Strongly clean power series rings, Proceedings of the Edinbergh Mathematical Society 50 (2007) 73–85.
https://doi.org/10.1017/S0013091505000404 - I.G. Connell, On the group ring, Canad. J. Math. 15 (1963) 650–685.
https://doi.org/10.4153/CJM-1963-067-0 - J. Han and W.K. Nicholson, Extensions of clean rings, Comm. Algebra 29 (2001) 2589–2596.
https://doi.org/10.1081/AGB-100002409 - N.A. Immormino and W.Wm. McGovern, Examples of clean commutative group rings, J. Algebra 405 (2014) 168–178.
https://doi.org/10.1016/j.jalgebra.2014.01.030 - D. Khurana and C. Kumar, Group rings that are additively generated by idempotents and units. arXiv:0904.0861 [math.RA]
- T.Y. Lam, A First Course in Noncommutative Rings (Springer-Verlag, Berlin, 2001).
- W. McGovern, A characterization of commutative clean rings, Int. J. Math. Game Theory Algebra 15 (4) (2006) 403–413.
- W.K. Nicholson, Lifting idempotents and exchange rings, Tran. Amer. Math. Soc. 229 (1977) 269–278.
https://doi.org/10.2307/1998510 - W.K. Nicholson, Local group rings, Canad. Math. Bull. 15 (1) (1972) 137–138.
https://doi.org/10.4153/CMB-1972-025-1 - W.K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasg. Math. J. 46 (2) (2004) 227–236.
https://doi.org/10.1017/S0017089504001727 - D. Passman, The Algebraic Structure of Group Rings (Dover Publications, 2011).
- Y. Qu and J. Wei, Abel rings and super-strongly clean rings, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), (2017), Tomul LXIII, f. 2, 265–272.
https://doi.org/10.1515/aicu-2015-0011 - X. Wang and H. You, Cleanness of the group ring of an Abelian p-group over a commutative rings, Alg. Colloq. 19 (3) (2012) 539–544.
https://doi.org/10.1142/S1005386712000405 - Y. Zhou, On clean group rings, Advances in Ring Theory, Trends in Mathematics (2010) 335–345.
https://doi.org/10.1007/978-3-0346-0286-0-22
Close