Strongly nil *-clean rings
Keywords:
Rings with involution, Strongly nil ∗-clean ring, ∗-Boolean ring, Boolean ring
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.
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M. Chacron, On a theorem of Herstein, Canad. J. Math. 21 (1969) 1348–1353.
H. Chen, On strongly J–clean rings, Comm. Algebra 38(10) (2010) 3790–3804.
H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011.
H. Chen, A. Harmancı A. Ç. Özcan, Strongly J–clean rings with involutions, Ring theory and its applications, Contemp. Math. 609 (2014) 33–44.
A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211.
A. L. Foster, The theory of Boolean–like rings, Trans. Amer. Math. Soc. 59 (1946) 166–187.
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.
C. Li, Y. Zhou, On strongly *–clean rings, J. Algebra Appl. 10(6) (2011) 1363–1370.
V. Swaminathan, Submaximal ideals in a Boolean–like rings, Math. Sem. Notes Kobe Univ. 10(2) (1982) 529–542.
L. Vaš, *–Clean rings; some clean and almost clean Baer *–rings and von Neumann algebras, J. Algebra 324(12) (2010) 3388–3400.
Published
2017-05-15
Section
Articles
