Strongly nil *-clean rings

Year 2017, Volume: 4 Issue: 2 (Special Issue: Noncommutative rings and their applications), 155 - 164, 11.01.2017

Abdullah Harmanci Huanyin Chen A. Cigdem Ozcan

https://doi.org/10.13069/jacodesmath.284954

Abstract

A $*$-ring $R$ is called {\em strongly nil $*$-clean} if every element of $R$ is the sum of a
projection and a nilpotent element that commute with each other.
In this paper we investigate some properties of strongly nil
$*$-rings and prove that $R$ is a strongly nil $*$-clean ring if
and only if every idempotent in $R$ is a projection, $R$ is
periodic, and $R/J(R)$ is Boolean. We also prove that a $*$-ring
$R$ is
commutative, strongly nil $*$-clean and every primary ideal is maximal if and only if every element of $R$ is a projection.

Keywords

Rings with involution, Boolean ring, Strongly nil *-clean ring, *-Boolean ring

References

• [1] A. Badawi, On abelian $\pi$–regular rings, Comm. Algebra 25(4) (1997) 1009–1021.
• [2] S. K. Berberian, Baer *–Rings, Springer-Verlag, Heidelberg, London, New York, 2011.
• [3] M. Chacron, On a theorem of Herstein, Canad. J. Math. 21 (1969) 1348–1353.
• [4] H. Chen, On strongly J–clean rings, Comm. Algebra 38(10) (2010) 3790–3804.
• [5] H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011.
• [6] H. Chen, A. Harmancı A. Ç. Özcan, Strongly J–clean rings with involutions, Ring theory and its applications, Contemp. Math. 609 (2014) 33–44.
• [7] A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211.
• [8] A. L. Foster, The theory of Boolean–like rings, Trans. Amer. Math. Soc. 59 (1946) 166–187.
• [9] Y. Hirano, H. Tominaga, A. Yaqub, On rings in which every element is uniquely expressable as a sum of a nilpotent element and a certain potent element, Math. J. Okayama Univ. 30 (1988) 33–40.
• [10] C. Li, Y. Zhou, On strongly *–clean rings, J. Algebra Appl. 10(6) (2011) 1363–1370.
• [11] V. Swaminathan, Submaximal ideals in a Boolean–like rings, Math. Sem. Notes Kobe Univ. 10(2) (1982) 529–542.
• [12] L. Vaš, *–Clean rings; some clean and almost clean Baer *–rings and von Neumann algebras, J. Algebra 324(12) (2010) 3388–3400.