Strongly $P$ -clean and semi-Boolean group rings

  • D. Udar Indian Inst. Technology, Delhi, India
  • R. K. Sharma Indian Inst. Technology, Delhi, India
  • J. B. Srivastava Indian Inst. Technology, Delhi, India
Keywords: Strongly P-clean rings, semiboolean rings, group rings, clean rings

Abstract

A ring $R$ is called clean (resp., uniquely clean) if every element is (uniquely represented as) the sum of an idempotent and a unit.
A ring $R$ is called strongly P-clean if every its element can be written as the sum of an idempotent and a strongly nilpotent element that commute.
The class of strongly P-clean rings is a subclass of classes of semi-Boolean and strongly nil clean rings.
A ring $R$ is called semi-Boolean if $R/J(R)$ is Boolean and idempotents lift modulo $J(R),$ where $J(R)$ denotes the Jacobson radical of $R.$
The class of semi-Boolean rings lies strictly between the classes of uniquely clean and clean rings.
We obtain a complete characterization of strongly P-clean group rings.
It is proved that the group ring $RG$ is strongly P-clean if and only if $R$ is strongly P-clean and $G$ is a locally finite 2-group. Further, we also study semi-Boolean group rings.
It is proved that if a group ring $RG$ is semi-Boolean, then $R$ is a semi-Boolean ring and $G$ is a 2-group and that the converse assertion is true if $G$ is locally finite and solvable, or an FC group.



References

Chen H., K¨ose H., Kurtulmaz Y. Strongly P-clean rings and matrices // Int. Electron. J. Algebra. – 2014. – 15. –

P. 116 – 131.

Chen J., Nicholson W. K., Zhou Y. Group rings in which every element is uniquely the sum of a unit and an

idempotent // J. Algebra. – 2006. – 306. – P. 453 – 460.

Connell I. G. On the group ring // Canad. J. Math. – 1963. – 15. – P. 650 – 685.

Diesl A. J. Nil clean rings // J. Algebra. – 2013. – 383. – P. 197 – 211.

ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12

D. UDAR, R. K. SHARMA, J. B. SRIVASTAVA

Ko¸san T., Wang Z., Zhou Y. Nil-clean and strongly nil-clean rings // J. Pure and Appl. Algebra. – 2016. – 220, № 2. –

P. 633 – 646.

Lam T. Y., First A. Course in noncommutative rings. – Second ed. – New York: Springer-Verlag, 2001.

McGovern W. Wm., Raja S., Sharp A. Commutative nil clean group rings // J. Algebra and Appl. – 2015. – 14, № 6. –

p.

Nicholson W. K. Local group rings // Canad. Math. Bull. – 1972. – 15, № 1. – P. 137 – 138.

Nicholson W. K. Lifting idempotent and exchange rings // Trans. Amer. Math. Soc. – 1977. – 229. – P. 269 – 278.

Nicholson W. K., Zhou Y. Rings in which elements are uniquely the sum of an idempotent and a unit // Glasg. Math.

J. – 2004. – 46. – P. 227 – 236.

Nicholson W. K., Zhou Y. Clean general rings // J. Algebra. – 2005. – 291. – P. 297 – 311.

Passman D. S. The algebraic structure of group rings. – New York: John Wiley and Sons, 1977.

Wang Z., Chen J. L. 2-Clean rings // Canad. Math. Bull. – 2009. – 52, № 1. – P. 145 – 153.

Zhou Y. On clean group rings // Adv. Ring Theory, Trends Math. – Basel: Birkh¨auser, 2010. – P. 335 – 345.

Received 07.06.16

Published
16.01.2020
How to Cite
Udar, D., R. K. Sharma, and J. B. Srivastava. “Strongly $P$ -Clean and Semi-Boolean Group Rings”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, no. 12, Jan. 2020, pp. 1717-22, https://umj.imath.kiev.ua/index.php/umj/article/view/2289.
Section
Short communications