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A NOTE ON STRONGLY *-CLEAN RINGS
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 Title & Authors
A NOTE ON STRONGLY *-CLEAN RINGS
CUI, JIAN; WANG, ZHOU;
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 Abstract
A *-ring R is called (strongly) *-clean if every element of R is the sum of a projection and a unit (which commute with each other). In this note, some properties of *-clean rings are considered. In particular, a new class of *-clean rings which called strongly -*-regular are introduced. It is shown that R is strongly -*-regular if and only if R is -regular and every idempotent of R is a projection if and only if R/J(R) is strongly regular with J(R) nil, and every idempotent of R/J(R) is lifted to a central projection of R. In addition, the stable range conditions of *-clean rings are discussed, and equivalent conditions among *-rings related to *-cleanness are obtained.
 Keywords
(strongly) *-clean ring;(strongly) clean ring;strongly -*-regular ring;stable range condition;
 Language
English
 Cited by
1.
Some characterizations of ∗-regular rings, Communications in Algebra, 2017, 45, 2, 841  crossref(new windwow)
 References
1.
A. Badawi, On abelian ${\pi}$-regular rings, Comm. Algebra 25 (1997), no. 4, 1009-1021. crossref(new window)

2.
S. K. Berberian, Baer *-Rings,, Grundlehren der Mathematischen Wissenschaften, vol. 195, Springer-Verlag, Berlin-Heidelberg-New York, 1972.

3.
K. P. S. Bhaskara Rao, The Theory of Generalized Inverses over Commutative Rings, Taylor & Francis, London and New York, 2002.

4.
G. Borooah, A. J. Diesl, and T. J. Dorsey, Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra 212 (2008), no. 1, 281-296. crossref(new window)

5.
V. P. Camillo and H. P. Yu, Exchange rings, units and idempotents, Comm. Algebra 22 (1994), no. 12, 4737-4749. crossref(new window)

6.
H. Chen, Rings with many idempotents, Int. J. Math. Math. Sci. 22 (1999), no. 3, 547-558. crossref(new window)

7.
J. Chen and J. Cui, Two questions of L. Vas on *-clean rings, Bull. Aust. Math. Soc. 88 (2013), no. 3, 499-505. crossref(new window)

8.
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), no. 2, 453-460. crossref(new window)

9.
J. Chen, X. Yang, and Y. Zhou, When is the $2{\times}2$2 matrix ring over a commutative local ring strongly clean? J. Algebra 301 (2006), no. 1, 280-293. crossref(new window)

10.
A. Y. M. Chin and H. V. Chen, On strongly ${\pi}$-regular group rings, Southeast Asian Bull. Math. 26 (2002), no. 3, 387-390.

11.
C. Li and Y. Zhou, On strongly ${\ast}$-clean rings, J. Algebra Appl. 10 (2011), no. 6, 1363-1370. crossref(new window)

12.
W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. crossref(new window)

13.
W. K. Nicholson, Strongly clean rings and Fitting's lemma, Comm. Algebra 27 (1999), no. 8, 3583-3592. crossref(new window)

14.
W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasg. Math. J. 46 (2004), no. 2, 227-236. crossref(new window)

15.
L. Vas, ${\ast}$-Clean rings: some clean and almost clean Baer ${\ast}$-rings and von Neumann algebras, J. Algebra 324 (2010), no. 12, 3388-3400. crossref(new window)

16.
Z. Wang, J. Chen, D. Khurana, and T. Y. Lam, Rings of idempotent stable rang one, Algebr. Represent. Theory 15 (2012), no. 1, 195-200. crossref(new window)