On \(*\)-\(r\)-Clean Rings
DOI:
https://doi.org/10.26713/cma.v16i3.3126Keywords:
Involution, ∗-r-clean ring, ∗-clean ring, Ring extensionsAbstract
A ring with an involution \(*\) is called \(*\)-\(r\)-clean if every element is a sum of a projection and a regular element (in the sense of von Neumann). In this article, several connections between \(*\)-\(r\)-clean rings and \(*\)-clean rings are given. Different characterizations of \(*\)-\(r\)-clean rings are presented. The passage of this property to some extensions of rings is studied.
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References
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics, Springer-Verlag, New York, x + 378 pages (1992).
N. Ashrafi and E. Nasibi, r-clean rings, Mathematical Reports 15(65) (2013), 125 – 132.
N. Ashrafi and E. Nasibi, Rings in which elements are the sum of an idempotent and a regular element, Bulletin of the Iranian Mathematical Society 39(3) (2013), 579 – 588.
S. K. Berberian, Baer ∗-rings, Die Grundlehren der Mathematischen Wissenschaften, Band 195, Springer-Verlag, New York – Berlin, xiii + 296 pages (1972).
B. Brown and N. H. McCoy, The maximal regular ideal of a ring, Proceedings of the American Mathematical Society 1 (1950), 165 – 171, DOI: 10.1090/S0002-9939-1950-0034757-7.
H. Chen, Rings with many idempotents, International Journal of Mathematics and Mathematical Sciences 22(3) (1999), 547 – 558, DOI: 10.1155/S0161171299225471.
J. Cui and Z. Wang, A note on strongly ∗-clean rings, Journal of the Korean Mathematical Society 52(4) (2015), 839 – 851, DOI: 10.4134/JKMS.2015.52.4.839.
D. Handelman, Perspectivity and cancellation in regular rings, Journal of Algebra 48(1) (1977), 1 – 16, DOI: 10.1016/0021-8693(77)90289-7.
P. Kanwar, A. Leroy and J. Matczuk, Idempotents in ring extensions, Journal of Algebra 389 (2013), 128 – 136, DOI: 10.1016/j.jalgebra.2013.05.010.
C. Li and Y. Zhou, On strongly ∗-clean rings, Journal of Algebra and Its Applications 10(6) (2011), 1363 – 1370, DOI: 10.1142/S0219498811005221.
W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Mathematical Journal 46(2) (2004), 227 – 236, DOI: 10.1017/S0017089504001727.
W. K. Nicholson, Lifting idempotents and exchange rings, Transactions of the American Mathematical Society 229 (1997), 269 – 278, DOI: 10.2307/1998510.
A. Sharma and D. Basnet, Weakly r-clean rings and weakly ⋆-clean rings, Analele Stiin¸tifice ale Universitatii “Alexandru Ioan Cuza” din Iasi. Matematica (Serie noua) 65(2) (2019), 227 – 239, URL: https://www.math.uaic.ro/~annalsmath/index.php/1271-2/.
L. Vaš, ∗-Clean rings; some clean and almost clean Baer ∗-rings and von Neumann algebras, Journal of Algebra 324(24) (2010), 3388 – 3400, DOI: 10.1016/j.jalgebra.2010.10.011.
Z. Wang, J. Chen, D. Khurana and T.-Y. Lam, Rings of idempotent stable range one, Algebras and Representation Theory 15 (2012), 195 – 200, DOI: 10.1007/s10468-011-9276-4.
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